Neutrosophic Super Matrices and Quasi Super Matrices by Florentin Smarandache

ZIP PUBLISHING Ohio 2012

Neutrosophic Super Matrices and Quasi Super Matrices W. B. Vasantha Kandasamy Florentin Smarandache

Thisbookcanbeorderedfrom:

ZipPublishing 1313ChesapeakeAve. Columbus,Ohio43212,USA TollFree:(614)485-0721 E-mail:info@zippublishing.com Website:www.zippublishing.com

Peerreviewers:

Prof.ValeriKroumov,OkayamaUniv.ofScience,Japan. Dr.S.Osman,MenofiaUniversity,ShebinElkom,Egypt. Dr.SebastianNicolaescu,2TerraceAve.,WestOrange,NJ07052,USA. Prof.GabrielTica,BailestiCollege,Bailesti,Jud.Dolj,Romania. Manybookscanbedownloadedfromthefollowing

ISBN-13: 978-1-59973-194-0 EAN: 9781599731940 Printed in the United States of America

DigitalLibraryofScience: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm

3 CONTENTS Preface 5 Chapter One INTRODUCTION 7 Chapter Two NEUTROSOPHIC SUPER MATRICES 9 Chapter Three SUPER BIMATRICES AND THEIR GENERALIZATION 87 3.1SuperBimatricesandtheirProperties87 3.2OperationsonSuperBivectors139

4 Chapter

157

Four QUASI SUPER MATRICES

FURTHER READING 195 INDEX 197 ABOUT THE AUTHORS 200

PREFACE

Inthisbookauthorsstudyneutrosophicsupermatrices.The conceptofneutrosophyorindeterminacyhappenstobeonethe powerfultoolsusedinapplicationslikeFCMsandNCMswhere theexpertseeksforaneutralsolution.Thusthisconcepthaslots ofapplicationsinfuzzyneutrosophicmodelslikeNRE,NAM etc.Theseconceptswillalsofindapplicationsinimage processingwheretheexpertseeksforaneutralsolution.

Hereweintroduceneutrosophicsupermatricesandshow thatthesumorproductoftwoneutrosophicmatricesisnotin generalaneutrosophicsupermatrix.

Anotherinterestingfeatureofthisbookisthatweintroduce anewclassofmatricescalledquasisupermatrices;these matricesarethelargerclasswhichcontainstheclassofsuper matrices.Theseclassofmatricesleadtomorepartitionofn×m matriceswheren>1andm>1,wheremandncanalsobe equal.Thusthisconceptcannotbedefinedonusualrow matricesorcolumnmatrices.

Thesematriceswillplayamajorrolewhenstudyinga problemwhichneedsmultifuzzyneutrosophicmodels.

Thisbookisorganizedintofourchapters.Chapteroneis introductoryinnature.Chaptertwointroducesthenotionof neutrosophicsupermatricesanddescribesoperationsonthem. Inchapterthreethemajorproductofsuperrowvectorsand columnvectorsaredefinedandareextendedtobisupervectors.

Thefinalchapterintroducesthenewconceptofquasisuper matricesandsuggestssomeopenproblems.

WethankDr.K.Kandasamyforproofreadingandbeing extremelysupportive.

W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE

INTRODUCTION

Inthischapterwejustindicatetheconceptswhichwehave usedandtheplaceofreferenceofthesame.

ThroughoutthisbookIwillbedenotetheindeterminate suchthatI2=I.

definitionandpropertiesofsupermatricespleaserefer[2,5]. Thenotionofbimatricesisintroducedin[6].

Hereinthisbookthenotionofneutrosophicsupermatrices areintroducedandoperationsonthemareperformed.

Wealsocanusetheentriesofthesesuperneutrosophic matricefromrings;(Z  I)(g)={a+bg|a,b

I andg isthenewelementsuchthatg2=0}.ZcanbereplacedbyRor QorZnorCandwecallthemasdualnumberrings[11].

Chapter One







Q 



















I and C(Zn)  I denotestheneutrosophicring[3-4,7].Forthe





Z 

8NeutrosophicSuperMatricesandQuasiSuperMatrices

(Q  I)(g1)={a+bg1|a,b  Q  I whereg1isanew elementsuchthat2 1 g=g1}.Wecall(Q  I)(g1)asspecial duallikenumberneutrosophicrings.HerealsoQcanbe replacedbyRorZorCorZn.

(R  I)(g2)={a+bg|a,b  R  I,g2thenewelement suchthat2g2=–g2}isthespecialquasidualnumber neutrosophicring.WecanreplaceRbyQorZorZnorCand stillwegetspecialquasidualnumberneutrosophicrings.

Finallywecanalsouse(Z  I)(g1,g2,g3)={a1+a2g1+ a3g2+a4g3|ai  Z  I;1  i  4;g1,g2,g3arenewelements suchthat2 1 g=0,2g2=g2and2 3 g=–g3withgigj=(0orgiorgj), i  j;1  i,j  3}thespecialmixedneutrosophicdualnumber rings[11-13].

Finallyinthisbookthenewnotionofquasisupermatrices isintroducedandtheirpropertiesarestudied.Thisnewconcept ofquasisupermatricescanbeusedintheconstructionoffuzzyneutrosophicmodels.

NEUTROSOPHIC SUPER MATRICES

Inthischapterweforthefirsttimeintroduceandstudythe notionofsuperneutrosophicmatricesandtheirproperties.

AsupermatrixAinwhichtheelementindeterminacyIis presentiscalledasuperneutrosophicmatrix.Tobemore precisewewilldefinethesenotions.

DEFINITION 2.1: Let X = {(x1 x2 … xr | xr+1 … xt | xt+1 …. xn) where xi   R  I ; R reals 1 i  n) be a super row matrix which will be known as a super row neutrosophic matrix.

Thusallsuperrowmatricesingeneralarenotsuperrow neutrosophicmatrix.

Weillustratethissituationbyanexample.

Example 2.1: LetX=(3I1–4|20–5|2I01|7214),Xisa superrowneutrosophicmatrix.

Allsuperrowmatricesarenotsuperrowneutrosophic matrices.

Chapter Two

FortakeA=(31011|25–27|01234),Aisonlya superrowmatrixandisnotasuperrowneutrosophicmatrix.

Example 2.2: LetA=(I3I1|–I5I|2I03I21|03);Aisa superrowneutrosophicmatrix.

Example 2.3: Let A=(0.3I,0,1|0.7,2I,0I+7|–8I,0.9I+2);Aisasuper rowneutrosophicmatrix.

DEFINITION 2.2: Let Y =

            

y y y y y

1 2 r t1 n

where yi  R  I; 1  i  n. Y is a super column matrix known as the super column neutrosophic matrix.

Weillustratethissituationbysomeexamples.

Example 2.4: Let

10NeutrosophicSuperMatricesandQuasiSuperMatrices



   1

Y= 2 I 34I

                  

0.3I

beasupercolumnmatrix.Yisclearlyaneutrosophicsuper columnmatrix(superneutrosophiccolumnmatrix).

Example 2.5: Let

beasuperneutrosophiccolumnvector/matrix.

Itispertinenttomentionherethatingeneralallsuper columnmatricesarenotsuperneutrosophiccolumnmatrices. Thisisprovedbyanexample.

Example 2.6: Let

NeutrosophicSuperMatrices11

Y= 3I 2 0 1 5 I 0.4 5 6I                               

3 0 1 31 4 0.5 2 1                           ,

beasupercolumnmatrixwhichisclearlynotasuper neutrosophiccolumnmatrix.

Wenowproceedontodefinethenotionofneutrosophic supersquarematrix.

DEFINITION 2.3: Let

aaa

be a n  n super square matrix where aij  R  I . A is called the neutrosophic super square matrix of order n  n. Weillustratethisbyanexample.

Example 2.7: Let

12NeutrosophicSuperMatricesandQuasiSuperMatrices

                     1112

n1 nn

= (aij) =

1n 2122 2n

aaa aa

A= 2I0572 I130I2I 01I3I7 250206 1353I7 111I1I                    

bea66supersquarematrix.ClearlyastheentriesofAare from R  I.Aisneutrosophicsupersquarematrix.

Example

Example 2.9: Let

Aisjustasquaresupermatrixandisnotasquareneutrosophic supermatrix. Nowweproceedontodefinearectangularsupermatrix.

NeutrosophicSuperMatrices13

5I60135I6I

                     

2.8: Let A= 30I01I5I I1015I01 23II00.323 121I0I2I 342I23I34

13I56789I

, Aisa77neutrosophicsquaresupermatrix. Allsupersquarematricesneednotingeneralbea neutrosophicsquaresupermatrices. Weillustratethisbyanexample.

A= 12345 67890 09876 54321 24807                

14NeutrosophicSuperMatricesandQuasiSuperMatrices DEFINITION

Let M = (mij) =                      111213 1n 212223 2n p11p12p13 p1n p1p2p3 pn mmmm mmmm mmmm mmmm be a p  n (p  n) rectangular super matrix where mij  R  I . Clearly M is a rectangular neutrosophic p  n super matrix. Weillustratethisbyanexample. Example 2.10: Let A= 30I654I32I1 520I23I45I6 1I50I23I45I I0110I1I2       Aisa49rectangularneutrosophicsupermatrix. Example 2.11: Let A= 3I120 I4I53I 123II4 4I5602I 78I937 012I89I                     bea65rectangularsuperneutrosophicmatrix.

2.4:

Allsuperrectangularmatricesingeneralneednotbea neutrosophicsuperrectangularmatrices.

Weillustratethisbyanexample.

2.12: Let

bea4  8superrectangularmatrix.ClearlyAisnotasuper neutrosophicrectangularmatrix.

Thuswecansayasincaseofsupermatricesifwetakeany neutrosophicmatrixandpartitionit,wewouldgetthesuper neutrosophicmatrix.

Wecanasincaseofsupermatricesdefineaneutrosophic supermatrixisoneinwhichoneormoreofitselementsare themselvessimpleneutrosophicmatrices.Theorderofthe neutrosophicsupermatrixisdefinedinthesamewayasthatof aneutrosophicmatrix.

Wehavetodefineadditionandsubtractionofneutrosophic supermatrices.

WecannotaddingeneraltwoneutrosophicmatricesAand Bofsamenaturalordersaym  n.

Weshallfirstillustratethisbyanexample.

NeutrosophicSuperMatrices15



  

Example

A= 12345678 90987654 32101234 56789012    





betwosuperneutrosophicmatricesofnaturalorder76. ClearlywecannotaddAwithBthoughassimplematricesA andBcanbeadded.

Thusfortwosupermatricestohaveacompatibleadditionit isnotsufficienttheyshouldbeofsamenaturalorderthey shouldnecessarilybeofsamenaturalorderbutsatisfyfurther conditions.Thisconditionweshalldefineasmatrixorderofa supermatrix.

DEFINITION 2.5: Let A =

be a neutrosophic super matrix where each aij is a simple neutrosophic matrix. 1  i  m and 1  j  n.

The natural order of A is t  s; clearly t  m and s  n that is A has t rows and s columns.

The matrix order of the neutrosophic super matrix A is m  n. i.e., it has m number of matrix rows and n number of

16NeutrosophicSuperMatricesandQuasiSuperMatrices Example 2.13: Let A= 356781 012I34 I0100I 1234I0 0I1234 01I301 0110I0            andB= 53I206 010I12 I0120I 0I60I0 I23405 0123I0 6I720I                      

             11 1n m1 mn aa aa

columns. The number rows in each simple neutrosophic matrix aip, i fixed is the same, p may vary; 1  p  n. Likewise for akj; j fixed, 1  k  m; the number columns is the same. Wenowgiveexamplesofthese.

NeutrosophicSuperMatrices17

Let A= 3I12I102I 02000I210 10I11I312 2113I2134 4I200I156I II01010I3I                     = 111213 212223 313233 aaa aaa aaa           wherea11= 3I1 020    ,a12= 2I10 00I2       ,a13= 2I 10       , a21= 10I 211 4I2      ,a22= 11I3 3I21 00I1           ,a23= 12 34 56I           , a31=[II0],a32=[1010]anda33=[I3I]aretheneutrosophic submatricesofAalsoknownasarethecomponentmatricesof A. ThenaturalorderofAis6  9andthematrixorderofAis 3  3.

matrix

Example 2.14:

DEFINITION 2.6: Let A and B be two neutrosophic super matrices of same natural order say m  n. Let A and B be a neutrosophic super matrices of matrix order t  s where t  m and s  n. We can say A+B the sum of the super neutrosophic matrices A and B be defined if and only if each simple matrix aij in A is of same order for each bij in B. 1  i  t; 1  j  s; i.e., if A =

bbb bbb where aij and bij are simple matrices. Here order of aij = order of bij; 1 i  s and 1  j  t. Weillustratethisbysomeexamples.

18NeutrosophicSuperMatricesandQuasiSuperMatrices

Example 2.15: Let A= 3I012 00I10 11123 I0456I 1I0I3         = 1112 2122 3132 aa aa aa           besuperneutrosophicmatrix.ThenaturalorderofAis55 andthematrixorderofAis32. Nowhavingseenexampleofmatrixorderofaneutrosophic supermatrixwenowproceedontodefineadditionoftwosuper neutrosophicmatrices.



    



1112 1s t1t2 ts aaa aaa and B =          



1112

t1t2 ts

NeutrosophicSuperMatrices19 Example 2.16: Let A=11121314 21222324 aaaa aaaa     wherea11= 3I1 101       a12= I 2       , a13= 10 01    ,a14= 111I 00I0       ,a21=[01I],a22=[3], a23=[1I]anda24=[51I0]. ThenaturalorderofAis310. A= 3I0I10111I 10120100I0 01I31I51I0           . ThematrixorderofAis24. TakeaneutrosophicsupermatrixBofnaturalorder310 andmatrixorder24givenby B= 01I010101I 10112101I0 I0121I1052      = 11121314 21222324 bbbb bbbb     Sincenaturalorderofmatrixaij=naturalorderofbij,for eachiandj,1  i  2and1  j  4andasAandBhavesame naturalorderaswellasmatrixorder.WecanaddAandB.

20NeutrosophicSuperMatricesandQuasiSuperMatrices A+B= 3I1II202122I 202322012I0 I151I522I615I2        .



Example 2.17: Let A= I0123I0 1I0321I 01I2340 501I0I1 2340I1I         = 111213 212223 aaa aaa         and B= 5I33I1I 01420I0 I32121I 1233410 0I15I1I         = 111213 212223 bbb bbb         betwosuperneutrosophicmatricesofnaturalorder5

3.ClearlyA+B,i.e.,thesumofAwithBisnot

11  b11,a12  b12,a21 

22 

WeseeA+Bisalsoaneutrosophicsupermatrixofnatural order3

10andmatrixorder2

4. ItisimportanttomentionthatevenifAandBaretwo neutrosophicsupermatricesofsamenaturalorderandsame matrixorderyetthesumofAandBmaynotbedefined. Thisisestablishedbyanexample.

7and matrixorder2

definedasa

b21anda

.Hencethe claim. Thuswecanrestatethedefinitionofthesumoftwo neutrosophicsupermatricesasfollows.

NeutrosophicSuperMatrices21



 j 

i.e., A =         11 1t s 1st aa aa and B =         11 1t s 1st bb bb where

Weillustratethisbyanexample. Example 2.18: Let A= 105I2 1234I I0123 67890        andB= 160I1 0102I I2110 01212               betwosuperneutrosophicmatricesofsamenaturalandmatrix order. Sinceeachofthecorrespondingcomponentmatricesare equaltheirsumisdefined.

DEFINITION 2.7: Let A and B be two super neutrosophic matrices of same natural order say m n and matrix order s t. The addition of A with B (B with A) is defined if and only if the natural order of each aij equal to natural order of bij for every 1

s and 1

aij and bij are simple matrices or the component submatrices of A and B respectively; 1  i  s and 1  j  t. Onsimilarlinesthesubtractionordifferencebetweentwo neutrosophicsupermatricescanbedefined. Anotherimportantfactabouttheadditionofneutrosophic supermatricesisthatsumoftwoneutrosophicsupermatrices neednotingeneralgivewaytoaneutrosophicsupermatrixit canbeonlyasupermatrix.

be a type A super vector where each Vi’s are column subvectors; 1  i  n. If some of the Vi’s are neutrosophic column vectors then we call V to be a type A neutrosophic super vector or neutrosophic super vector of type A. Weillustratethisbysomeexamples. Example 2.19: Let

22NeutrosophicSuperMatricesandQuasiSuperMatrices

              .

             1 2 n

V= 3 I 0 1 1 0 4 I                           = 1 2 3 V V V           ,

NowA+B= 26503 13360 02233 6810102

WeseeA+Bisonlyasupermatrixandisnota neutrosophicsupermatrix. Nowwecanhavethefollowingtypeofsuperneutrosophic matrices. DEFINITION 2.8: Let

V V V

VisaneutrosophicsupervectoroftypeA.

VisaagainaneutrosophicsupervectoroftypeA.Allsuper vectorsoftypeAneednotbeneutrosophicsupervectoroftype A.

NeutrosophicSuperMatrices23

V= I 2 1 0 1 1 I 1 I                               = 1 2 3 4 V V V V



Example 2.20: Let

 

  

     ,

V= 1 2 3 4 5 6 9 8 0 1                                 = 1 2 3 V V V          

Weillustratethisbyanexample. Example 2.21: Let

24NeutrosophicSuperMatricesandQuasiSuperMatrices

V= 3 0 1 4 1 2 9 8 7                               = 1 2 3 V V V          

3 V V V     

   

beasupervectoroftypeA.ClearlyVisnotaneutrosophic supervectoroftypeA. Example 2.22: Let

beasupervectoroftypeAandisnotaneutrosophic supervector. LetV= 1 2

beasupervectoroftypeA. ThetransposeofVisagainasupervectorwhichisarow supervectorgivenby Vt=[3014|12|978]=ttt 123 VVV

. WenowdefineneutrosophicsupervectoroftypeB. DEFINITION 2.9: Let T = [a11 | a12 | … | a1n] be a neutrosophic super vector where aij is a m mi matrix 1  i  n. T is a neutrosophic super vector of type B. We call the transpose of this neutrosophic super vector of type B to be also a neutrosophic super vector of type B.

Weillustratethisbythefollowingexamples.

AisaneutrosophicsupervectoroftypeB.

NeutrosophicSuperMatrices25

1 2 3 a

  

     

021 5I0 112 I01 013 0I0 103 114 I25                   





A= 21356 I0123 4I567 00I01 10023 0I450 70311 3011I I6100 025I0 1I011 23612 3I013                                           = 1 2 3 A A A          

Example 2.23: Let A=

a a



    



 , AisaneutrosophicsupervectoroftypeB. Example 2.24: Let

WewishtosaysupervectoroftypeBareingeneralnot neutrosophic.

AisonlyasupervectoroftypeBwhichisnota neutrosophicsupervectoroftypeB. Wedefinethetransposeofaneutrosophicsupervectorof typeB.

26NeutrosophicSuperMatricesandQuasiSuperMatrices



                  



Weillustratethissituationbyanexample. Example 2.25: Let A= 315 021 311 402 116 789 123 456 789



  

   = 1 2 3 A A A           ,





DEFINITION 2.10: Let A =            1 n A A be a neutrosophic super vector of type B. The transpose of A denoted by At =     

    

 t 1 2 ttt 12n b A A AAA A

each Ai is a neutrosophic matrix which has same number of columns. A is also a neutrosophic super vector of type B.

NeutrosophicSuperMatrices27

Let A= 30I 123 987 760 894 140 271 390 52I 123                                 = 1 2 3 A A A           beaneutrosophicsupervectoroftypeB. At = t 1 ttt 2123 3 A AAAA A           = 3197812351 0286947922 I3704010I3           isagainaneutrosophicsupervectoroftypeB.

here

Weillustratethisbythefollowingexample. Example 2.26:

Asincaseofneutrosophicsupermatriceswecandefinethe sumoftwoneutrosophicsupervectorsoftypeAandtypeB.

DEFINITION 2.11: Let A =

                  

1 1 m n1 n

  

a A A a a be a neutrosophic supervector of type A.

Let the natural order of A be n  1.

Suppose B =

b B B b

1 1 m n

   

                  

B is also a neutrosophic super vector of type A of same natural order say n  1.

The sum of A and B denoted by A+B is defined if and only if the natural order of Ai is equal to the natural order of Bi for every i, i = 1,2,…, m i.e., both the matrices A and B have same matrix order.

Weillustratethisbythefollowingexample.

28NeutrosophicSuperMatricesandQuasiSuperMatrices

NeutrosophicSuperMatrices29



0 1 I 0 2 3 I



     + I 0 1 1 1 0 2                         = I 0 I1 1 3 3 2I                           .

Example 2.27: Let A= 1 2 3 0 1 IA 0A 2A 3 I         

             andB= 1 2 3 I 0 1B 1B 1B 0 2         

             betwoneutrosophicsupermatricesoftypeA. A+B=



    

ClearlyA+BisalsoaneutrosophicsupervectoroftypeAsame naturalorderandmatrixorderthatofAandB. Wecandefinealsothesubtractionofneutrosophicsuper vectoroftypeA.NowA–Biscalculated.

Howeverwewishtostatethatsumofdifferenceoftwo neutrosophicsupervectoroftypeAneednotalwaysgivewayto neutrosophicsupervectoroftypeAitcanalsobeonlya supervectoroftypeA.

Weillustratethissituationbyanexample.

30NeutrosophicSuperMatricesandQuasiSuperMatrices A–B= 0 1 I 0 2 3 I             –I 0 1 1 1 0 2                         = I 1 I1 1 1 3 I2                         .

WeseeA–BisalsoaneutrosophicsupervectoroftypeAof samenaturalorderandmatrixorderasthatofAandB.

Example 2.28: Let A= I 1 0 2 1 I 0 2 3                               andB= I 1 0 1 4 I 2 1 0                              

betwoneutrosophicsupervectoroftypeA.

NeutrosophicSuperMatrices31 A+B= I 1 0 2 1 I 0 2 3                               + I 1 0 1 4 I 2 1 0                               = 0 2 0 3 5 0 2 3 3                               isonlyasupervectoroftypeAandnotaneutrosophicsuper vectoroftypeA. Wejustindicatebyanexamplehowtheminorproductof typeAneutrosophicsupervectorsiscarriedout. Example 2.29: Let A=[3I0|1214|0I][A1A2A3]and B= 1 2 3 0 1 I 0B 1B 2B 0 2 0                          .

NowingeneralminorproductofAandB,twoneutrosophic supervectorsoftypeAmaynotbedefined.IfAisthe neutrosophicsuperrowvectoroftypeAwithA=[A1…At] withnaturalorder1  nandmatrixorder1  tthenifBisa neutrosophiccolumnsupervectoroftypeBthentheminor productofAwithB=[B1…Bt]tisdefinedifandonlyifnatural orderofBisn  1andthematrixorderofBist  1withnatural orderofeachBiinBisthesameasthetransposeofthenatural orderofeachAiinAfori=1,2,…,torequivalentlythe naturalorderofAiinAisequaltotheorderofthetransposeof thenaturalorderofBiinBforeachi=1,2,…,t.

32NeutrosophicSuperMatricesandQuasiSuperMatrices

0 1 I 0 1 2 0 2 0             

              =[3I0]

    

           



  

TheminorproductofAB=[3I0|1214|0I]



0 1 I

+[1214]

0 2 1 0 +[0I]



 0 2 ={30+I1+0I}+{10+21+12+40} +{02+I0} ={0+I+0}+{0+2+2+0}+[0+0] =I+4+0=I+4.

Thusweseetheminorproductofaneutrosophicsuperrow vectorAwiththetransposeoftheAiswelldefinedforallthe conditionsrequiredforthecompatibilityoftheminorproductof neutrosophicsupervectorsoftypeAaresatisfied.

Wewillillustratethissituationbysomeexamplesbeforewe

NeutrosophicSuperMatrices33

Let A=[10I2|001|23I01]



ClearlyAt = 1 0 I 2 0 0 1 2 3 I 0 1                                      

proceedontodefinemoretypesofproductsonthese neutrosophicsupervectorsofdifferenttypes. Example 2.30:

beaneutrosophicsupervectoroftypeAofnaturalorder1  12 andmatrixorder1

isalsoaneutrosophicsupervectoroftypeAforwhichAAt,the matrixminorproductiswelldefined.

34NeutrosophicSuperMatricesandQuasiSuperMatrices AAt=[10I2|001|23I01] 1 0 I 2 0 0 1 2 3 I 0 1                                       =[10I2] 1 0 I 2       +[001]           1 0 0 +[23I01] 2 3 I 0 1                 ={11+00+II+22}+{00+00+11}+ {22+33+II+00+11} ={1+0+I+4}+{0+0+1}+{4+9+I+0+1} ={5+I}+{1}+{14+I} =20+2I. Thuswehavethefollowingtheoremstheproofofwhichis leftasanexerciseforthereader.

THEOREM 2.1: Let A = [A1 A2 … At] be a neutrosophic super row vector of natural order 1  n and matrix order 1  t. Then AAt the minor product of AAt exists (well defined).

THEOREM 2.2: Let A be a neutrosophic super row vector of type A. If B denotes the matrix A without partition then AAt = BBt. NowforthesametypeAneutrosophicsupervectorAwe definethemajorproduct.

DEFINITION 2.12: Let A =

A A and B = [B1 … Bs] be two neutrosophic super vectors of type A. The major product of A with B is defined by AB =

Bs]

NeutrosophicSuperMatrices35

         

          

=             st stst s s BABABA BABABA BABABA      22122 21111 .



1 t A A [B1 …

Here each Ai and Bj are matrices for 1  i  t and 1  j  s. Weillustratethissituationbythefollowingexample.

36NeutrosophicSuperMatricesandQuasiSuperMatrices Example 2.31: Let A= 3 2 I 2 0 I 4             andB=[260|1I] betwoneutrosophicsupervectorsoftypeA. AB= 3 2 I 2 0 I 4                         [260|1I] =    33 226021I II 22 2601I 00 II 2601I 44                        

NeutrosophicSuperMatrices37 = 618033I 412022I 2I6I0II 412022I 00000 2I6I0II 824044I                       Example 2.32: Let A= 0 1 I 2 3 4 1 I 2 3                                 and B=[01|I20|340|I0]beanytwoneutrosophicsuper vectorsoftypeA. WeobtainthemajorproductofthesupercolumnvectorA withthesuperrowvectorBofthetwoneutrosophicmatrices.

38NeutrosophicSuperMatricesandQuasiSuperMatrices AB= 0 1 I 2 3 4 1 I 2 3                 [01|I20|340|I0] =    0000 1011I2013401I0 IIII 2222 3333 01I20340I0 4444 1111 IIII 2012I2023402I0 3300                                   

0000000000

01I20340I0 0II2I03I4I0I0 022I406802I0 033I6091203I0 044I80121604I0 01I20340I0 0II2I03I4I0I0 022I406802I0 033I6000000

               

beanytwoneutrosophicsupervectorsoftypeA.

NeutrosophicSuperMatrices39

A= 2 3 I 1 2 0 0 1 1                              

MajorproductoftwoneutrosophicsupervectorsoftypeA isillustratedbelowbythefollowingexample. Example 2.33: Let

andB=[301|0I1]

40NeutrosophicSuperMatricesandQuasiSuperMatrices BA=   210 301330123011 I01 210 0I130I120I11 I01                               = 32031I310210300111 02I31I01I21000I111      = 60I300001 03II02I00I1      = 6I31 4I2I1I         . Nowweproceedontodefinetheminorproductof neutrosophicsupervectorsoftypeB. Weonlyillustratethisbynumericalexamplesforthereader tounderstandtheminorproductwithoutanyconfusionasitwill involvelotofnotationsandsymbolsrepellingnon mathematiciansawayfromusingthesematrices. Example 2.34: LetA= 21301I 000100 I0I001      andB= 01 I2 10 31 10 21                     beanytwo neutrosophicsupervectorsoftypeB.

NeutrosophicSuperMatrices41 TheminorproductofABisdefinedasfollows: AB= 21301I 000100 I0I001           01 I2 10 31 10 21                     =  21301I31 01 0001010010 I2 I0I00121          = I43012II 000031 0II021             = 43I4I 31 2I1I               . Wefromthenumericalillustrationobservethefollowing facts. 1.Theminorproductsofneutrosophicsupervectors oftypeBisalwaysasimplematrix.Itmaybea simpleneutrosophicmatrixormaynotbeasimple neutrosophicmatrix. 2.IfA=[A1…At]andB= 1 t B B            areneutrosophic supervectorsoftypeB,theminorproductABis definedifandonlyif

(a)thenumberofsubmatricesinAandBareequal.

(b)TheproductAiBiiscompatible,i.e.,definedfor eachi,1  i  t.

(c)Theresultantsimplematrixisofordermn wheremisthenumberofrowsofAiandnis thenumberofcolumnsofBiwhichisthesame foreachAiandeachBii.e.,numberofrowsof AisthesameasthatofAiandthenumberof columnsofBisthesameasthatofeachBi. Wegiveyetanothernumericalillustrationtomakeone understandtheworking. Example 2.35:

42NeutrosophicSuperMatricesandQuasiSuperMatrices

LetA= 00I110210 1011I0012 I10001201      and B= 201I3 11000 00I10 12011 0123I 12001 10I10 20101 31011                             betwoneutrosophicsupervectorsoftypeB.

WenowfindtheminorproductofAwithB.

NeutrosophicSuperMatrices43 AB= 00I110210 1011I0012 I10001201      201I3 11000 00I10 12011 0123I 12001 10I10 20101 31011                             201I3 00I110 110000123I 1011I0 00I1012001 I10001 12011                   21010I10 01220101 20131011                 = 12I1I10123I 321II240I2I3II 2I11II3I12001             402I121 82123 512I31             = 533I32I6I2 114I3I24I4I7 2I743II33I2               .

44NeutrosophicSuperMatricesandQuasiSuperMatrices

2.36: Let A= 512411234 01011I123 103200I00 200I01010 I15010101         beaneutrosophicsupervectoroftypeB.Wenowfindthe

t . AAt = 512411234 01011I123 103200I00 200I01010 I15010101         5012I 11001 20305 412I0 11001 1I010 21I01 32010 43001                             = 51241 0105012I11 412I0 1031100120 11001 20020305I0 I1501                    

ThusweseeifAisaneutrosophicsupervectoroftypeB. thetransposeofAisdenotedbyAtthenAAtisjust neutrosophicsimplematrixorasimplematrix. Wewillillustratethisbyanexample. Example

minorproductofAA

NeutrosophicSuperMatrices45 1234 1I010 I123 21I01 0I00 32010 1010 43001 0101                      = 30111105I1117584I1 11001522I1 110102I158242I0 00242I4II2II0 5I111I152II2611001                             3020I2I4614I I2014III24 2III0I 4I2020 64I02              = 7726I192I144I5I18 I2617I2I2I26 192I2I14I22I2I15 144I2I222I6I2I 5I1862I152II29              . WeseeAAtisasymmetricsimpleneutrosophicmatrix.It isalwaysasquaresimplematrix.Thissimplematrixmaybe neutrosophicormaynotbeneutrosophicbutitisalways symmetric. Wegiveyetanothernumericalillustration.

46NeutrosophicSuperMatricesandQuasiSuperMatrices Example 2.37: LetAbeaneutrosophicsupervectoroftypeBgivenby A= 410213I10521012 2010120010I0600 53I301210101I10 1I010300I001001       AAt = 410213I10521012 2010120010I0600 53I301210101I10 1I010300I001001             4251 103I 01I0 2031 1100 3213 I020 1010 010I 5010 2I00 1011 06I0 1010 2001                        

NeutrosophicSuperMatrices47 =  410 2042511 01I0 53103II 1I0           213I2031 01201100 30123213 1030I020              1010 10521012 010I06I0 010I0600 50101010 10101I10 2I002001 0I001001 1011                          = 178234I0000 8410201I0 23103453I0II0 4I253I1I0000                 17892I11 7526 92I2146 116610                312I715012 2I1I0I0366I0 703116I1I0 1I11I2001                      

48NeutrosophicSuperMatricesandQuasiSuperMatrices = 67I152I402I18I 152I11I127I8I 402I127I522I123I 18I8I123I132I           . AAtisclearlyaneutrosophicsquaresymmetricmatrixof order4. Nowweproceedontoillustratemajorproductof neutrosophicsupervectorsoftypeBbyanexample. Example 2.38: Let A= 310 121 I01 012 60I 102 310 I00              andB= 10130121 21011002 0I10010I           betwoneutrosophicsupervectorsoftypeB.Tofind AB= 310 121 I01 012 60I 102 310 I00              10130121 21011002 0I10010I     

NeutrosophicSuperMatrices49  10130121 310310310 21011002 121121121 0I10010I 10130121 I0121I010I0111002 0I10010I 012 60I10 10221 3100I I00                         012012 13012160I60I 011002102102 10010I310310 I00I00                                  = 513101365 52I252224I III13I0I12I2I 212I2112022I 6I6I1806I126I 12I3303212I 513101365 I0I3I0I2II                                . ABisclearlyaneutrosophicsupermatrix.Itmaybeasuper matrixnotnecessarilyneutrosophic. Nowweillustratethemajorproductofneutrosophicsuper vectoroftypeBbyanexample.

50NeutrosophicSuperMatricesandQuasiSuperMatrices Example 2.39: Considerthetwoneutrosophicsupervectors. A= 2101 0110 1I00 I010 001I 1I00 000I 1000 0001               andB= 10010010I 0I000I000 001000010 0000I0001             ThemajorproductoftheneutrosophicsupervectorAwiththe neutrosophicsupervectorBisdescribedinthefollowing: AB= 2101 0110 1I00 I010 001I 1I00 000I 1000 0001               10010010I 0I000I000 001000010 0000I0001            

NeutrosophicSuperMatrices51 10010010I 210121012101 0I000I000 011001100110 001000010 0I000I000I00 0000I0001 10010 I0100I0I01000 001I001001I00 0000I                 010I I010I000 001I0010 0001 1I001001I00101I00010I 000I0I0000I00000II000 100000110000010000010 000100000010I00010001               

0000I000I 10010010I

                                  = 2I02II202I1 0I100I010 0I000I000 I01I00I1I 0010I001I 1I010I10I

0000I0001                . WeseeABisalsoaneutrosophicsupermatrix. Nowweillustratebyexamplesthemajorproductofa neutrosophicsupervectoroftypeBwithitstranspose.

52NeutrosophicSuperMatricesandQuasiSuperMatrices Example 2.40: Let A= 301 0I0 100 100 00I 001 100 010 I01 0I0                                 beaneutrosophicsupervectoroftypeB. NowthetransposeofAdenotedby At = 30110010I0 0I0000010I 1000I10010      AAt = 301 0I0 100 100 00I 001 100 010 I01 0I0                 30110010I0 0I0000010I 1000I10010           =

NeutrosophicSuperMatrices53 30130130110301010I0 0I00I00I0000I00010I 1001001000I10010010 30110010I0 100100100 0I0000010I 00I00I00I 1000I10              010 001001001 30110010I0100100100 0I0000010I010010010 1000I10010I01I01I01 0I00I00I0                                      =



10033I1303I10 0I0000010I 30110000I0 30110010I0 I000II00I0 1000I10010 30110010I0 0I0000010I 3I10III1I0I10 0I00000I0I

                 . WeseeinthefirstplaceAATisalsoaneutrosophicsuper matrix.FurtherAAthappenstobeasymmetricneutrosophic supermatrix. Thusasincaseofusualmatriceswegetincaseof neutrosophicrowsupervectorAoftypeB,AAThappenstobea symmetricneutrosophicsupermatrix.

Weillustratethisyetbyanexampleaswearemore interestedinthereadertoapplyanduseit,thatiswhyweare avoidingthenotationaldifficultwayofexpressingitasa definition. Example 2.41: Let

54NeutrosophicSuperMatricesandQuasiSuperMatrices

A= 01010010 00I01101 000110I0 I0000110      

At = 000I 1000 0I00 1010 0110 0101 10I1 0100                           .

beaneutrosophicsupervectoroftypeB.

NeutrosophicSuperMatrices55 Nowwefind AtA= 000I 1000 0I00 1010 0110 0101 10I1 0100              01010010 00I01101 000110I0 I0000110             000I000I 01010010 10001000 00I01101 0I000I00 000110I0 10101010 I0000110 01100110 01010 01010101 00I01 10I110I1 00011 01000100 I0000                           010 101 0I0 110                                                = I0000II0 01010010 00I0II0I 0102101I0 00I121I1 I0I01211 I101II12I0 00I01101                . Weseeclearlytheneutrosophicsupermatrixissymmetric.

56NeutrosophicSuperMatricesandQuasiSuperMatrices Wegetyetanotherexample. Example 2.42: LetB= 30110 0001I 10001 0I000 2000I 00100 01100 10010                           beaneutrosophicsupervectoroftypeB. Bt = 30102001 000I0010 10000110 11000001 0I10I000                 Consider BBt = 30110 0001I 10001 0I000 2000I 00100 01100 10010              30102001 000I0010 10000110 11000001 0I10I000                

NeutrosophicSuperMatrices57  3010 3011030110000I 0001I0001I1000 10001100011100 0I10 3010 000I 0I0000I0001000 1100 0I10 301 000 2000I 100 00100 110 0I1                                                              0 I 2000I 0 00100 0 0 3010 000I 0110001100 1000 1001010010 1100 0I10                                                                                  



 

 

                                                    111306114 11II0I001 3I202I001 000I00I0 6I2I04I002 10000110 100I0120 41102002

                       

      .

58NeutrosophicSuperMatricesandQuasiSuperMatrices



  

2001 30110301100010 0001I0001I0110 10001100010001 I000 2001 0010 0I0000I0000110 0001 I000 20 00 2000I 01 00100 00 I0  

                                                                 

01 10 2000I 10 00100 01 00 2001 0010 0110001100 0110 1001010010 0001 I000  =

NeutrosophicSuperMatrices59 ItiseasilyverifiedBB

Example 2.43: LetA= 20 0I 00 10 11 01           andB= 0110101 0000000       beaneutrosophicsupervectorsoftypeB. NowAB= 20 0I 00 10 11 01                     0110101 0000000       = 20012010101 0I000I00000 0000 10011010101 11001100000 0101                       

Tisagainaneutrosophicsupermatrix whichissymmetric.

WeseeABisasupermatrix.ClearlyABisnotaneutrosophic supermatrix.

Thusitispertinenttomentionherethatthemajorproduct oftwoneutrosophicsupervectorsoftypeBneednotingeneral beaneutrosophicsupermatrix;itcanonlybeasupermatrix.

Thisisseenfromexample.

Nowweproceedontoshowbyanumericalexamplehow theminorproductoftwosuperneutrosophicmatriceswhichhas acompatiblemultiplicationiscarriedout.

60NeutrosophicSuperMatricesandQuasiSuperMatrices = 0220202 0000000 0000000 0110101 0110101 0000000           .

Example 2.44: LetAandBbeanytwosuperneutrosophicmatrices,where A= 10I010 21010I I10001 011100 001010 0000I0 301101            B= 1001000I0 010001010 100II0I00 100001100 010100010 001010101                    

NeutrosophicSuperMatrices61 AB= 10I010 21010I I10001 011100 001010 0000I0 301101            1001000I0 010001010 100II0I00 100001100 010100010 001010101                     1 2 I 0 0 0 3                          [1001|00|0I0]+ 0I 10 10 11010001010 100II0I00 01 00 01                 

 

62NeutrosophicSuperMatricesandQuasiSuperMatrices 010 10I 001100001100 100010100010 010001010101 0I0 101                       =        111 1001000I0 222 III 000 1001000I0 000 000 3100130030I0                           

  +   0I01000I010I010 10100I10I010I00 101010 110100110111010 01100I01I001I00 000000 010001010 010101 100II0I00                              

NeutrosophicSuperMatrices63 100001100 010010010 010100010 10I10I10I 001010101 001001001 100001 100100100 010100 010010010 001010 0I00I00I0                   100 010 101 100001100 101010110100101010 001010101                                                      1001000I0I00II0I00 20020002I0010001010 I00I000I0010001010 000000000110II1I10 000000000100II0I00 000000000000000000 30030003I0100II0I00                        010100010 10I0I11I0I 001010101 100001100 010100010 0I0I000I0 101011201             

64NeutrosophicSuperMatricesandQuasiSuperMatrices = 1I102II0I1I0 31I2I21I2I1I I11I1111I1 210II21I10 1101II0I10 0I0I000I0 5013I1I12I3I1                  . Thustheminorproductisagainasuperneutrosophic matrix. Wegiveyetanotherillustration. Example 2.45: LetAandBbetwoneutrosophicsupermatriceswhere A= 1011200 0I0010I I121010 00I0101 1I01I00 00100I0 11201I1            and B= 011I00I 0110101 I001000 10I1001 0100100 1100I0I 0000011                      

NeutrosophicSuperMatrices65 AB= 1011200 0I0010I I121010 00I0101 1I01I00 00100I0 11201I1            011I00I 0110101 I001000 10I1001 0100100 1100I0I 0000011                       =  101 0I0 I12 000110101 011I00II 1II001000 0 001 112                              + 1200 010I 101010I1001 01010100100 1I001100I0I 00I00000011 01I1                       

66NeutrosophicSuperMatricesandQuasiSuperMatrices Heretheusualproductofeachsubmatricesareperformed andweseeinthisminorproductofneutrosophicsupermatrices eachcelliscompatiblewithusualproductofsubmatrices. =    111 011I001 000 III 000 011I001 111 000 10111I001                                +   0101011010011 I0I0I00100I00 121212 0I010I10100I1 I0I0I00100I00 010101 0110101 121212 I001000                                      +

NeutrosophicSuperMatrices67 10I1001 1200011200001012000 010I11010I00I0010II 0000011 1010101010I100 01010101010010 1I00111I0000I0 00I00000I00001                 10101 01010 1I00I 00I01 10I1001 0100100 01I101I101I1 1100I0I 0000011                              



 = 011I00II001000 00000000II0I0I 0III00I2I112101 0000000I00I000 011I00I0II0I0I 0000000I001000 011I00I2I112101                        12I1201 01001II 21I1I01I 0100111 1II1I01 II00I0I I1I001I11I                          

  



minorproductofneutrosophicmatrices.

Nowweshowtheminorproductofaneutrosophicsuper matrixwithitsspecialtransposecanbedefined. Letusfirstillustratebyafewexamplesthespecial transposeofasuperneutrosophicmatrix.

68NeutrosophicSuperMatricesandQuasiSuperMatrices =

112I12II12I02I1

          

 

1I31I2I201I 01II01II2I 22II22I1I31I022I I10I111

2II01I0I 3I3I22I2I122I

 

 . WeseeABisagainaneutrosophicsupermatrixunderthe

Example 2.46: LetA= 21312I0 1I01011 01I1102 3I10101 0101010 I110I00                    

beaneutrosophicsupermatrix.

TofindthetransposeoftheneutrosophicsupermatrixAwe takethetransposeofeachofthesubmatricesinA.

NeutrosophicSuperMatrices69

NowthespecialtransposeofAdenotedbyAtisgivenbya neutrosophicmatrix. At = 21030I 1I1I11 30I101 111010. 20110I I10010 012100                       Example 2.47: LetA= 21000303210 0152II1012I 7890712I012 0I010I60120 10I01211230 001001II127 10011021023 0I10I1I0I12 10102311101               beasuperneutrosophicmatrix.

70NeutrosophicSuperMatricesandQuasiSuperMatrices

At =

                

207010101 118I000I0 0590I1011 020100100 0I70101I2 3I1I21013 01261I2I1 30I01I101 2101210I1 121232210 0I2007321

clearlyAtisalsoaneutrosophicsupermatrix.

AtwillbeknownasthespecialtransposeofA.

;

ClearlyifAisaneutrosophicrowsupervectoroftypeB thenthespecialtransposeofA,Atisaneutrosophiccolumn supervectoroftypeB.

SimilarlyifAisaneutrosophiccolumnsupervectoroftype BtheAtthespecialtransposeofAisaneutrosophicrowsuper vectoroftypeB.

Weillustratethisbyafewexamples.

Example 2.48:

LetA=[210|111I0|03I0]beaneutrosophicsuper rowvectorthen

At =

                  

2 1 0 1 1 1 I 0 0 3 I 0

Example 2.49:

,Atisaneutrosophicsupercolumnvector.

LetA=

                

3 6I 1 2 0 1 1 I 0 2 I

aneutrosophicsupercolumnvector.

ThenthespecialtransposeofA;At=[36I120|11I0|2I].

ClearlyAtisaneutrosophicsuperrowvector.

NeutrosophicSuperMatrices71

72NeutrosophicSuperMatricesandQuasiSuperMatrices Example

3101I312I 121040123I I0I234I201 0143I01101      

At = 31I0 1201 01I4 1023I I430 304I1 1121 2200 I3I11                            

2.50

LetA=

beaneutrosophicsuperrowvectoroftypeB. ThenAtthespecialtransposeofA. ThatisifA=[A1A2A3]weseeAt = ttt 123 AAA     andthe rowvectorsarearrangedascolumnvectorssothatthenumber ofcolumnsofAisequaltothenumberofrowsofAt .

isneutrosophicsupercolumnvectoroftypeB.

NeutrosophicSuperMatrices73 Example 2.51: Let A= 0I120 100I0 1010I 23450 010I2 10015I 7I0213 01011 11111 IIII3 20410 5123I                                       beaneutrosophicsupercolumnvectoroftypeB.Thespecial transposeofAis At = 0112017I01I25 I00310011I01 101400201I42 2I05I1111I13 00I025I31130I         . WeseeAtisaneutrosophicsuperrowvectoroftypeB. Wenowillustratetheminorproductoftheneutrosophic supermatricesAwithitstranspose.

74NeutrosophicSuperMatricesandQuasiSuperMatrices

3011001201

41021I0012

0111210201

            

At = 301400I1 0I011102 10100100 10I23101 01010210 010I010I 10I00010 2010I200 0101001I 10121100                                

Example 2.52: LetA=

0I00110010 101I00I101

0103000I01

I000101010 12010I00I0

beaneutrosophicsupermatrix. TofindtheminorproductofAwithAt

NeutrosophicSuperMatrices75 AA

                         

                               

0I001 101I0 301400I1 4102110I23101 0I011102 0103001010210 10100100

                                     

t = 3011001201 0I00110010 101I00I101 41021I0012 0103000I01 0111210201 I000101010 12010I00I0

301400I1 0I011102 10100100 10I23101 01010210 010I010I 10I00010 2010I200 0101001I 10121100

= 30110

01112 I0001 12010

76NeutrosophicSuperMatricesandQuasiSuperMatrices + 01201 10010 010I010I 0I101 10I00010 I0012 2010I200 00I01 0101001I 10201 10121100 01010 I00I0                               301301301 301400I1 0I00I00I0 0I011102 101101101 10100100 410410410 301400I1 0100I010101010102 10100100 011 I00 120              301401100110I1 0I01I001I00102 10101200120100                       

NeutrosophicSuperMatrices77  





101 210                              010I010I 012010120101201 10I00010 100101001010010 2010I200 0I1010I1010I101 0101001I I0012I0012I0012 10121100 010I 10I0 00I01201          010I 0010 00I0100I01 0I200 0101001I 10121100 010I0 102011020110201 10I00 01010010100 2010I I00I0I00I001010 10121                      10I 010 1010200 I00I001I 100                                              

101010 0110I201301101 I00101I00I0210 212121 10I23101 303030 01010210 121212 10I23 010101 01010 101010             



 

78NeutrosophicSuperMatricesandQuasiSuperMatrices

                   = 17072I1642I73I14 03I022II3I24I 72I042I62I4I14I2I1I 1622I62II279I74I282I . 42II4I1911I52I05 73I4II752I1323I 3I122I4I2023I2I 44I1I82I53I2I62I                      

100412013I310I23101 0I0III02I01010210 402401I1I0I2I3II0I 12I417114I6212I56412 0I011102303I69303 1I11120212I43521 3I0I4I00II01010210 32I1622I510I23101                            603I22I1510 0201I0112I I302I21I3I0 21I2I52I212I 2I10I121I2I100 513I22I160I 11I1002I 02I02I0II2I 

ItisclearthatAAtisaneutrosophicsupersymmetric matrix.

ThusminorproductofamatrixAwithitsspecialtranspose Atgivesasupersymmetricmatrix.

ItispertinenttoobserveAisnotasymmetricneutrosophic matrix.

HenceAtisalsonotaneutrosophicsymmetricsupermatrix howevertheproductisasymmetricsuperneutrosophicsuper matrix.

Wegiveyetanotherexample.

Example 2.53:

beasupermatrix.

TofindtheminorproductofAwith

NeutrosophicSuperMatrices79

LetA= 30I04000101 0100010010I 60000010001 71011000011 10010101100 00110010110 1010I00I000 01I00I00I00 400010I0001 100I0011000 02110I00100 1000I010001                   

At =

                

306710104101 010100010020 I000011I0010 000111000I10 400100I0100I 0100100I00I0 00100100I101 000010I00100 1100110I0010 000101000000 1I1100001001

productofAwithAt .

AAt =

                  

30I04000101 0100010010I 60000010001 71011000011 10010101100 00110010110 1010I00I000 01I00I00I00 400010I0001 100I0011000 02110I00100 1000I010001

80NeutrosophicSuperMatricesandQuasiSuperMatrices

306710104101 010100010020 I000011I0010 000111000I10 400100I0100I 0100100I00I0 00100100I101 000010I00100 1100110I0010 000101000000 1I1100001001

                        

                  301 010 600 710 100 306710104101 001 010100010020 101 I000011I0010 01I 400 100 021 100

NeutrosophicSuperMatrices81

82NeutrosophicSuperMatricesandQuasiSuperMatrices 0400 0010 0001 1100 1010000111000I10 1001400100I0100I 0I000100100I00I0 00I000100100I101 010I I001 10I0 0I01                          0101 010I 0001 0011 1100000010I00100 01101100110I0010 I000000101000000 0I001I1100001000 0001 1000 0100 0000                          

NeutrosophicSuperMatrices83 301306301710103014101 010010010100010100020 6001006000011I6000010 710710 306 100100 010 001001 100 101101 01I01I                    710 710104101 100 100010020 001 0011I0010 101 01I 400400400 306710104 100100100 01010001 021021021 1000011I 100100100                          101 0020 0010                                  000111000I10 040004000400 400100I0100I 001000100010 0100100I00I0 000100010001 00100100I101 1100 000 1010 400 1001 010 0I00 001 00I0               11001100 111000I10 10101010 100I0100I 10011001 0100I00I0 0I000I00 00100I101 00I000I0 010I000010I I001400I0 10I0010 0I01001                     11100010I0I10 01100I0I001100I 10I00100I10I000I0 0I01001000I01I101                                   

84NeutrosophicSuperMatricesandQuasiSuperMatrices 000010I00100 010101010101 1100110I0010 010I010I010I 000101000000 000100010001 1I1100001000 0011 000 1100 110 0110 000 I000 1I1 0I00               00110011 010I00100 11001100 0110I0010 01100110 101000000 I000I000 100001000 0I000I00 00010000001 100011010 0100000 00001I1     

= 1001821314I12313 010100010020 1803642606024606 2114250707128727 306710104101 1000011I0010 4067112I4111 I1010II1I002I0 1202428404016404 306710104101 12020112I0050 306710104101                          +

               010I000010100 000110I10000010 01001010001000000 00001000000001000                            

NeutrosophicSuperMatrices85

0I00I00I00I0

4I01I01I02I102I                            + 21I11110I1010 1I2III110II010 1I1100001000 1I1201001000 110021II0110 1101120I0010 0000I0I00I00 II00II0I00I0 1I1100001000 000010I00100 1100110I0010 000000000000                      =

16004004I04004I 0100100I00I0 00100100I101 400211I01I1I 0101210I0I1I0 00111200I1I11 4I00I00I0I00I

40I10II01II02I 001II1I00I1II1 0I011I10I0I2I0

WeseetheminorproductofaneutrosophicsupermatrixA withitstransposeAtisasymmetricneutrosophicsupermatrix.

Theentriesinthesesupermatricescanbereplacedfromthe dualnumberneutrosophicrings,specialduallikenumber neutrosophicrings,specialquasidualnumberneutrosophic ringsandspecialmixeddualnumberneutrosophicrings.

86NeutrosophicSuperMatricesandQuasiSuperMatrices

           

           

281I19264244I2I173234I 1I22II1I2102I1I03I0 19I3843616025I707 261I4354827I1307I37I 4268521I2I42I2I1 21122512II1I31 44I067I1I122II4I1I131I 2I2I1012I2II13I0023I0 17I25I304I4I018I4I 

042I 3077I2I1I1I04I3II2 23I032I31323I0I62I0 34I077I111I042I2012I           

Chapter Three

SUPER BIMATRICES AND THEIR GENERALIZATION

Inthischapterweintroducethenotionofsuperbimatricesor equivalentlywillbeknownasbisupermatrices.Thestudyof bimatriceshasbeencarriedoutin[6].Herewedefinesuper bimatricesandgeneralizethem.

Thischapterhastwosections.Sectiononedefinessuper bimatricesandenumeratesafewoftheirproperties.Insection twothegeneralizationofsuperbimatrices(bisupermatrices)to supern-matrices(n-supermatrices)iscarriedout(n>2)when n=2thisstructurewillbeknownasbisupermatricesor superbimatrices.Whenn=3wecallitastrisupermatrixor supertrimatrix.Thiscanbemadeintoaneutrosophic superbimatrixbytakingtheentriesfrom

3.1 Super bimatrices and their properties

Inthissectionweintroducethenewnotionofsuper bimatrices(bisupermatrices)andenumerateafewofits algebraicpropertiesandoperationsonthem.

Z  I or Q  I

R  I or C  I or Zn  I.

DEFINITION 3.1.1 : Let A = A1  A2 where A1 and A2 are super row vectors of type A, A1 = (a1, a2, …, ar | ar+1… | … | am … an) and A2 = (b1, …, bs | bs+1… | … | bt … br). Then we call A = A1  A2 to be a super birow vectors (bisuper row vectors or super row bivectors) of type A if and only if A1 and A2 are distinct in atleast one of the partitions.

Weillustratethissituationbysomeexamples.

Example 3.1.1: LetA=A1  A2=(301|1104|–5|7831 2)  (10|312|1104|610).Aisasuperrowbivector (bisuperrowvector)oftypeA.

Example 3.1.2: LetA=A1  A2=(00|000|…|000)  (00000|00000|00),AisasuperrowbivectoroftypeA alsoknownasthesuperrowzerobivectoroftypeA.

Example 3.1.3: LetA=A1  A2=(1111|111111|11)  (11|1111|1111111);Aisasuperrowbivectorof typeA.InfactAisalsoknowasthesuperrowunitbivectorof typeAorbisuperrowunitvectoroftypeA.

Example 3.1.4 : LetA=A1  A2=(310|45|11125)  (301|45|11125);AisnotasuperrowbivectorforA1and A2arenotdistinct.IfA=A1  A2=(3|1045|11125)  (30|1451|11125)thenAisasuperrowbivectoroftype A.

Nowweproceedontodefinethenewnotionofbilengthof thesuperrowbivectoroftypeA.

DEFINITION 3.1.2: Let A = A1  A2 = (a1 … | …. | …| ar … an)  (b1 … | …. | …| bs … bn) be a super row bivector of type A where both A1 and A2 are of length n but they are distinct in the partitions. We say A is of bilength n and denote the bilength by (n, n).

88NeutrosophicSuperMatricesandQuasiSuperMatrices

If A = A1  A2 = (a1 … | …. | …| ar … an)  (b1 … | …. | …| bs … bn) be a super row bivector of type A (m  n) then we say A is of bilength (m, n).

Weillustratethisbysomeexamples.

Example 3.1.5: LetA=A1  A2=(1023|56789|21)  (0120|10927|53)beasuperbirowvectoroftypeA.We sayAisofbilength(11,11),i.e.,thissuperbirowvectorhas samebilength.

Example 3.1.6: Let A=A1  A2=(010|253|110)  (3120|21|5) beasuperbirowvectoroftypeA.Aisofbilength(9,7).We seethissuperrowbivectorhasdifferentbilengthforits componentsuperrowvectorsA1andA2.

Wecandefinethenotionofsuperrowbivectorequivalently inthisform.

DEFINITION 3.1.3: Let A = A1  A2, if A1 and A2 are two distinct super row vectors of type A, then we call A to be a super row bivector of type A.

Nowweproceedontodefinethenotionofsupercolumn bivectoroftypeA(superbicolumnvectororbisupercolumn vectoroftypeA).

DEFINITION 3.1.4: Let A = A1  A2 where A1 and A2 are super column vectors of type A which are distinct. Then we say A is a super column bivector of type A or bisuper column vector of type A or super bicolumn vector of type A.

Weillustratethissituationbythefollowingexamples.

SuperBimatricesandtheirGeneralization89

Example 3.1.7: Let

A=A1  A2=

                                         

5 3 7 1 7 0 2 1 4 1 0 1 3 0 1 7 2

beasuperbicolumnvectoroftypeA.ClearlyA1andA2are distinct.

Example 3.1.8: Let

A=A1  A2=

                                                       

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

beasupercolumnbivectoroftypeA.WesayAisthezero supercolumnbivectoroftypeAorsupercolumnzerovectorof typeA.

90NeutrosophicSuperMatricesandQuasiSuperMatrices

Example 3.1.9:

AisasupercolumnbivectoroftypeAknownasthesuper columnunitbivectoroftypeA.

DEFINITION 3.1.5: Let A = A1  A2 =

ab ab be a super column bivector of type A. If (m  n) then we say A is of super bilength (m, n).

                    

   

11 mn

If in A = A1  A2 where A1 and A2 are distinct super column vectors, if the length of A1 is the same as that of A2 then we say A is a super column bivector of length n and denote the bilength by (n, n).

SuperBimatricesandtheirGeneralization91

Let A= 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1                                                   =A1  A2.

Example 3.1.10:Let

A=A1  A2=

                                  

32 01 20 56 15 37 08 19 11 04 40

beasupercolumnbivectoroftypeA.Aisofbilength(11,11), i.e.,bothA1andA2areoflength11.ClearlyA1andA2are distinctsupercolumnvectors.

Example 3.1.11:Let

A=A1  A2=

92NeutrosophicSuperMatricesandQuasiSuperMatrices

1 4 00 1 2 3 7 1 5 2 8 1 9 1 1 510 7

 

 

  



 





               



 



            

beasupercolumnbivectoroftypeA.WesayAisofbilength (11,10).ClearlyA1andA2areofdifferentlengths.

Thesenewtypeofsuperbirowvectorsandsuperbicolumn vectorswillbeusefulincomputernetworking,incoding/ cryptographyandinfuzzymodels.

Nowweproceedontodefinethetransposeofasuperrow bivectorandsupercolumnbivectoroftypeA.

DEFINITION 3.1.6: Let A = A1  A2 be a super row bivector of type A. We define the transpose of A as the transpose of each of the super row vectors A1 and A2, i.e.,

At = (A1  A2)t = AAtt 21  . Clearly the transpose of a super row bivector is a super column bivector of type A.

Likewise if A = A1  A2 is a super column bivector of type A then the transpose of A is a super row bivector of type A, i.e., At is the union of the transpose of the column vectors A1 and A2 respectively. We denote At = (A1  A2)t = AAtt 21  .

Weshallillustratethisbythefollowingexample.

beasuperrowbivectoroftypeA.

SuperBimatricesandtheirGeneralization93

Example 3.1.12: LetA=A1  A2 =(0123|431|703568)  (30|2541|00725)

94NeutrosophicSuperMatricesandQuasiSuperMatrices

A1  A2= 31 03 10 25 56 67 18 09 33 71 42 55 94 

  



 



ThetransposeofAdenotedbyAt = 0 13 20 32 45 34 11 70 00 37 52 65 8                                                         = tt 12AA  . ItisclearthatAtisasupercolumnbivectororsuper bicolumnvectoroftypeA. Example 3.1.13: LetA=





      

beasupercolumnbivectoroftypeA.ThetransposeofA denotedbyAt = tt 12AA  =[301256|103|7459]  [13 0|56789|3–1254].

ItiseasilyverifiedthatAtisasuperrowbivector.

Nowhavingseenthetransposewhenistheadditionand multiplicationofthesesuperbivectorsoftypeAispossible.

DEFINITION 3.1.7: Let A = A1  A2 and B = B1  B2 be two super row (column) bivectors of type A. We say A and B are similar or “identical in structure” (identical in structure does not mean they are identical) if the following conditions are satisfied.

1. The bilength of A and the bilength of B are the same i.e., if bilength of A = A1  A2 is (m,n) then the bilength of B = B1  B2 is also (m,n).

2. The partition of the row (column) vector Ai is identical with that of the row (column) vector Bi, true for i=1,2. Weillustratethisbythefollowingexamples. Example 3.1.14:LetA=A1  A2 =[3156|023|15–2315]  [80|123|7|5678] beasuperrowbivector. B=[0401|6–12|301107]

SuperBimatricesandtheirGeneralization95

 [35|010|8|3015] beanothersuperrowbivector.WecansayAandBare identicalinstructure. WeseebilengthofAis(13,10)andthatofBis(13,10), theyareequal.

ClearlythesupercolumnbivectorsAandBareidenticalin structureandweseethebilengthofAis(9,10)andthatofBis (9,10)i.e.,thebilengthofAandBarethesame.

ThusweseeiftwosuperbivectorsoftypeAareidenticalin structuretheyalsoenjoythepropertythattheyhavesame bilength.Onlytomakethingsclearwementionitasaseparate propertythoughtheconceptidenticalinstructurecoversthe equalbilength.

Howeveritispertinenttomentionherethatequalbilength willnotingeneralimplytheyareidenticalinstructure.But identicalinstructurewillalwaysimplytheyhavesamebilength.

Nowweproceedontodefinethenotionofadditionofsuper bivectorsoftypeA.WhenwesaysuperbivectorsoftypeAit impliesthesuperbivectorcanbeasubbirowvectoroftypeA orsuperbicolumnvectoroftypeA;i.e.,thetermsuperbivector oftypeAincludesbothrowbivectoraswellascolumn bivector.

96NeutrosophicSuperMatricesandQuasiSuperMatrices

A1  A2= 8 3 1 0 3 1 4 5 5 7 7 8 1 1 2 8 0 0 3                                                andB= 0 1 2 1 1 1 2 0 3 1 4 2 5 3 9 4 8 5 7                

Example 3.1.15: LetA=

                               =B1  B2.

DEFINITION 3.1.8: Let A = A1  A2 and B = B1  B2 be any two super row bivectors which are identical in structure. Then addition or subtraction of A with B denoted by A + B = (A1  A2)  (B1  B2) is well defined and Ai  Bi is carried out component wise and A  B is again a super birow vector which are identical in structure with A and B.

Onsimilarlineswecandefinetheadditionorsubtractionof AwithBincaseofsupercolumnbivectorsAandBwhichare identicalinstructure.

SuperBimatricesandtheirGeneralization97

A= 3 2 0 1 1 0 2 3 5 4 7 7 3 2 4 0 5                             =A1  A

4 0 1 1 1 2 2 3 3 1 0 0 1 2 4 5 1         

 

     

 

      =B

Weillustratethisbythefollowingexamples. Example 3.1.16: Let

2andB=



1  B2 betwosupercolumnbivectorswhichareidenticalinstructure. WefindA+BthesumofAwithBtobe(A1+B1)  (A2+B2)

98NeutrosophicSuperMatricesandQuasiSuperMatrices = 7 2 1 0 0 2 4 6 8 3 7 7 4 4 8 5 4      



1–B1  A2–B2 = 1 2 1 2 2 2 0 0 2 5 7 7 2 0 0 5 6



    

                           =A+B. ClearlyA+Bisagainasupercolumnbivectorwhichare identicalinstructurewithAandB. NowA–B=A

                                         . WeseeA–BisagainsuperrowbivectoroftypeA identicalinstructurewithAandB.Thusweseethepropertyof identicalinstructureispreservedunderbothadditionand subtraction. Inviewofthiswehavethefollowinginterestingresult.

THEOREM 3.1.1: Let S = {A = A1  A2 / the collection of all super row bivectors of type A which are identical in structures with A having its entries from Q or R or C or Zn or C(Zn) or Z(g), Z(g1) or Z(g2) (g2 = 0 is a new element 2 g1 = g1 is a new element and 2 2 g = –g2 a new element n is such that, 2  n  }. Then S is a group under the operation of addition.

Theproofisleftasanexerciseforthereader.

THEOREM 3.1.2: Let B = {B = B1  B2 / the collection of all super column bivectors of type A which are identical in structure of type A with B having its entries from Q or R or C or Zn or C(Zn) or Z(g) or Z(g1) or Z(g2) (g2 = 0 is a new element 2 g1 = g1 is a new element and 2 2 g = –g2 a new element; n is such that, 2  n  }. Then B is a group under the operation of addition of the super column bivectors.

Thisproofisalsosimpleleftasanexerciseforthereader.

SuperBimatricesandtheirGeneralization99

Nowwedefinetheproductofasuperrowbivectoroftype AwithasupercolumnbivectoroftypeAwherethe multiplicationiscompatible. Wefirstshowhowtheminorproductoftwosupervectors aredefinedwheneverthemultiplicationhappenstobe compatible. Thisisillustratedbysomeexamples. Example 3.1.17: Letx=[301|2345]andy=[100|2316]t . Theminorproductofthesupervectorsxwithyisgivenby

100NeutrosophicSuperMatricesandQuasiSuperMatrices xy=[301|2345] 1 0 0 2 3 1 6                       =[301]           0 0 1 +[2345]             6 1 3 2 =3+{4+9+4+30} =3+47=50. Example 3.1.18:Let x=[0123|45|7890123|8]andy= 6 1 2 3 4 5 7 8 9 6 1 2 3 8                                            

SuperBimatricesandtheirGeneralization101 betwosupervectors xy=[0123|45|7890123|8] 6 1 2 3 4 5 7 8 9 6 1 2 3 8                                             =[0123] 6 1 2 3       +[45] 4 5       +[7890123] 7 8 9 6 1 2 3                       +[8][–8] ={0–1+4+9}+{–16+25}+{49–64+81+0–1+4–9}+{–64} =12+9+60–64=17.

102NeutrosophicSuperMatricesandQuasiSuperMatrices

Note: ThecompatabilityofeachViwithWiiswelldefinedfor i=1,2,…,n,thisisillustratedbythefollowingexamples. Example 3.1.19: Let x=[12015|320|70–12348|8014] beasuperrowvector.

DEFINITION 3.1.9: Let x = [V1 V2 … Vn] and y =             Wn W W  2 1 be two super vectors of type A. xy = [V1 V2 … Vn]             n W W W  2 1 = V1 W1 + V2 W2 + … + Vn Wn.

SuperBimatricesandtheirGeneralization103 t 1 2 0 1 5 3 2 0 7 x0 1 2 3 4 8 8 0 1 4                                                              xxt=[12015]                 5 1 0 2 1 +[320]           0 2 3 +

104NeutrosophicSuperMatricesandQuasiSuperMatrices [70–12348]                       8 4 3 2 1 0 7 +[8014]             4 1 0 8 ={1+4+0+1+25}+{9+4+0}+ {49+0+1+4+9+16+64}+{64+0+1+16} ={31+13+143+81} =268. Thuswecandefinetheproductincaseofsuperrowvector xwithxtwhichisasfollows: Ifx=[V1V2…Vn]thenxt = t 1 t 2 t n V V V                . Nowxxt=[V1V2…Vn] t 1 t 2 t n V V V                . ttt 1122nn VVVV...VV  .

SuperBimatricesandtheirGeneralization105

x=[01|3401|–3216789|32145]then t 0 1 3 4 0 1 3 2 1 x 6 7 8 9 3 2 1 4 5                                                         

Weillustratethisbyanexample. Example 3.1.20: Let

wherextisatransposeofxandxisasuperrowvector.

106NeutrosophicSuperMatricesandQuasiSuperMatrices t 0 1 3 4 0 1 3 2 1 xx[01|3401|3216789|32|145]6 7 8 9 3 2 1 4 5                                                          =[01]       1 0 +[3401]             1 0 4 3 +[–3216789]                       9 8 7 6 1 2 3

SuperBimatricesandtheirGeneralization107 +[32145]                 5 4 1 2 3 ={0+1}+{9+16+0+1}+{9+4+1+36+49 +64+81}+{9+4+1+16+25} =1+26+244+55 =326. Nowweproceedontodefinethenotionofmajorproductof

Weshallonlyillustratethisbysimpleexamples. Example 3.1.21:Let x= 0 1 2 4 5 7 8 1 4               andy=[12|0345|89210] betwosupervectorsoftypeA. Themajorproductofxwithydenotedby

twosupervectors.

108NeutrosophicSuperMatricesandQuasiSuperMatrices xy= 0 1 2 4 5 7 8 1 4                             [12|0345|89210] =    000 11210345189210 222 444 555 12034589210 777 888 111 12034589210 444                                             00000000000 12034589210 24068101618420 4801216203236840 510015202540451050 714021283556631470 816024324072811890 12034589210 4801216203236840               

                         

SuperBimatricesandtheirGeneralization109

4 7

Themajorproductofxwithydenotedxy=

Example 3.1.22:Let 1 4 3 7 0 2 5 1 xandy[340|21013|1214|1] 0 1 1 2 3

betwosupervector.

                        

110NeutrosophicSuperMatricesandQuasiSuperMatrices  1 4 3 7 0 2 5 1 0340|21013|1214|1 1 1 2 3 4 7 0

Weseeformajorproducttobedefinedweneednothave (1)Thenaturalorderofthesupercolumnvectortobeequal tothetransposeofthesuperrowvector. (2)Thenumberofpartitionsofxneednotbeequaltothe numberofpartitionsofy.





000 111 111 222 02101312141 333 444 777 000

                                                   

SuperBimatricesandtheirGeneralization111  

0 1 1 2 34 3 4 7 0



1111 3402101312141 4444 3333 7777 0000 3402101312141 2222 5555 1111

                    

 3402101312141 12160840412484164 912063039363123 2125014707217147287 0000000000000 6804202624282 1520010505155105205 3402101312141 0000000000000 3402101312141 3402101312141 6804202624282 91206303936312  3 12160840412484164 2128014707217147287 0000000000000                                                  

 

                       

112NeutrosophicSuperMatricesandQuasiSuperMatrices

matrix. Nowweshowbyexamplesthestructureofamajorproduct ofasupercolumnvectorwithitstranspose. Example 3.1.23: Let A= 3 0 1 5 1 2 3 7 0 2 1 5 0 1                                             beasupercolumnvector.TofindthemajorproductofA withAt . HereAt=[3015|–1237021|501]

Clearlythemajorproductoftwosupervectorsisasuper

SuperBimatricesandtheirGeneralization113 NowAAt = 3 0 1 5 1 2 3 7 0 2 1 5 0 1                                             [3015|–1237021|501]    333 000 30151237021501 111 555 111 222 333 30151237021501 777 000 222 111 55 030150 11                                                       5 12370210501 1                                                     

 50525510153501052505 00000000000000 30151237021501

                     

90315369210631503 00000000000000 30151237021501 150525510153501052505 30151237021501 60210246140421002 90315369210631503 210735714214901473500 00000000000000 60210246140421002 30151237021501 1

WeseethemajorproductofAwithitstransposeisa symmetricsupermatrix.

Thesymmetryinthemoccursinaveryspecialwaythe maindiagonalcomponentisasymmetricmatrixwhereasthe othercomponentsaretransposeofeachother;thiscanbe observedfromtheabovesupermatrix.

114NeutrosophicSuperMatricesandQuasiSuperMatrices

SuperBimatricesandtheirGeneralization115 Example 3.1.24:Let A= 3 1 2 0 1 4 1 0 2 3 4 5 6                                           bethesupercolumnvector. TofindthemajorproductofAwithitstranspose. At=[3|12|014|1023456]. WeseethemajorproductofAandAtresultsinaspecial symmetricsupermatrixwhosenaturalorderaswellasthe matrixorderisasquarematrix.

116NeutrosophicSuperMatricesandQuasiSuperMatrices WefindAAt.AAt = 3 1 2 0 1 4 1 0 2 3 4 5 6                                           [3|12|014|1023456]. =            33312301431023456 1111 3120141023456 2222 0000 13112101411023456 4444 111 000 222 312 333 444 555 666                      1 0 2 0141023456 3 4 5 6                                 

93603123069121518 3120141023456 624028204681012 0000000000000 3120141023456 1248041640812162024 3120141023456 0000000000000 624028204681012 93603123069121518 1248041640812162024 155100520501015202530 186120624601218243036 ItiseasilyobservedthatAAtisasupersymmetricmatrix. Thistypeofminorandmajorproductsincaseofsuper vectorsareextendedtosuperbivectorswhichwillbeexhibited byoneortwoexamples. Example 3.1.25: Let A=[0121|34|021]  [100011|012|41]and

                    

SuperBimatricesandtheirGeneralization117

118NeutrosophicSuperMatricesandQuasiSuperMatrices B= 0 11 21 31 40 00 11 12 03 10 1                                                  betwosuperbivectors.TheminorproductofA=A1  A2with B=B1  B2isgivenby AB=(A1  A2)(B1  B2) =A1B1  A2B2 =[0121|34|021] 1 2 3 4 0 1 1 0 1                              [100011|012|41] 0 1 1 1 0 0 1 2 3 0 1                                  

SuperBimatricesandtheirGeneralization119 =  1 1 20 0121340210 31 1 4                              0 1 1 10 100011012241 11 3 0 0                                  ={12+4+1}  {0+8+1} ={17}  {9}. Wegiveyetanotherexample. Example 3.1.26: Let A=[01234|56|7|8901234]  [012|01|3015] and

betwosuperbivectors.

                                                    

0 1 0 11 50 10 05 20 12 01 10 02 2 1 0

TofindtheminorproductofAwithB.

(Incaseofminorproductweseeeachofthecomponent submatrixoftherowsupermatrixismultipliedbythe correspondingcomponentsubmatrixofthecolumnsupermatrix wegettheresultanttobeasingletonbimatrix,whichisclearly notasupermatrix).

120NeutrosophicSuperMatricesandQuasiSuperMatrices

SuperBimatricesandtheirGeneralization121

AB=[01234|56|7|8901234] 0 1 0 1 5 1 0 2 1 0 1 0 2 1 0                                                  [012|01|3015] 1 0 0 5 0 2 1 0 2                            

122NeutrosophicSuperMatricesandQuasiSuperMatrices =  1 00 11 1 0123456728901234 00 0 12 51 0                                                2 1 51 0120013015 00 0 2                             ={24+5+14+9}  {0+0+16}={52}  {16}. Nowweproceedontoillustratethemajorproductoftwo superbivectorsinananalogouswayaswehavedoneforsuper vectorbysomeexamples. Example 3.1.27:LetA= 1 0 0 1 1 2 2 3 3 4 1 0 4 1 5 5 6 0 7 1 0 0 1 1 0                                                              

SuperBimatricesandtheirGeneralization123 andB=[10|2431|0123010]  [3120|11200|30| 251011]betwosuperbivectors.Tofindthemajorproduct ofthesetwosuperbivectors. AB=(A1  A2)(B1  B2) =A1B1  A2B2 = 0 1 2 3 1 4 5 6 7 0 1                                   [10|2431|0123010]  1 0 1 2 3 4 0 1 5 0 1 0 1 0                                             [3120|11200|30|251011]

                     



124NeutrosophicSuperMatricesandQuasiSuperMatrices       000 1011243110123010 222 333 0124310123010 111 444 555 666 0124310123010 777 000 111          

 



 

  

 

  

 



    1111 31201120030251011 0000 1111 2222 31201120030251011 3333 4444 0000 131201112001301251011 5555 0                              000 1111 31201120030251011 0000 1111                                  













                    

SuperBimatricesandtheirGeneralization125 = 0000000000000 0124310123010 0248620246020 03612930369030 0124310123010 0481612404812040 051020155051015050 061224186061218060 071428217071421070 0000000000000 0124310123010                   31201120030251011 00000000000000000 31201120030251011 624022400604102022 936033600906153033 12480448001208204044 00000000000000000 31201120030251011 15510055100015010255055 00000000000000000 31201120030251011 00000000000000000 31201120030251011                      Thusthemajorproductoftwosuperbivectorsisasuper bimatrix. Nowwegiveonemorenumericalillustrationofthemajor productoftwosuperbivectors

                                                                   

126NeutrosophicSuperMatricesandQuasiSuperMatrices Example 3.1.28:Let 3 1 1 02 1 3 1 0 1 1 2 0 A01 = 00 02 63 1 2 01 0 5 2 1

=A1  A2andB=B1  B2=[3100|120567|210]  [1 01|23405|110002103]betwosuperbivectorstofind themajorproductofAwithB. Incaseofthemajorproductofacolumnsuperbimatrixwith arowrowsuperbimatrixwegettheresultanttobeasuper bimatrixrectangularsquare. AB=(A1  A2)(B1  B2)=A1B1  A2B2

SuperBimatricesandtheirGeneralization127 3 1 1 0 2 1 3 1 0 1 1 2 0 0 [3100|120567|210][101|23405|110002103] 1 0 0 0 2 6 3 1 2 0 1 0 5 2 1                                                    333 1310011205671210 000 111 111 3100120567210 111 222 000 000 000 3100120567210 666 111 000 000 22 3100 11                                                            2 120567210 1                         

128NeutrosophicSuperMatricesandQuasiSuperMatrices      111 222 333 10123405110002103 000 111 000 111 000 21012234052110002103 333 222 101234051100021 111                          

   03 51015234055110002103                      9300360151821630 3100120567210 0000000000000 3100120567210 3100120567210 3100120567210 6200240101214420 0000000000000 0000000000000 0000000000000 1860061203036421260 3100120567210 0000000000000 0000000000000 62002  40101214420 3100120567210                          

            

SuperBimatricesandtheirGeneralization129

WeseeABtheminorproductofthetwosuperbivectorsisa superbimatrix. Nowweproceedontoillustratethemajorproductofa supercolumnvectorwithitstranspose,bysomenumerical examples. Weseethemajorproductofacolumnsupervectorwithits transposewhichisarowsupervectorgettheresultanttobea squaresymmetricsupermatrix.

10123405110002103 202468010220004206 3036912015330006309 00000000000000000 10123405110002103 00000000000000000 10123405110002103 00000000000000000 202468010220006206 3036912015330006309 202468010220004206 10123405110002103 50510152002555000105015                     

130NeutrosophicSuperMatricesandQuasiSuperMatrices Example 3.1.29

A= 2 1 0 1 3 5 0 1 0 3 2 1 7                                           beasupercolumnvector.TofindthemajorproductofAwith itstransposeAt=[2101|35010|321|7]. AAt = 2 1 0 1 3 5 0 1 0 3 2 1 7                      [2101|35010|321|7]

:Let

SuperBimatricesandtheirGeneralization131    2222 1111 2101350103217 0000 1111 3333 5555 2101350103217 0000 1111 0000 33 221012 11                                                        33 35010232127 11 721017350107321[7][7]                                  = 420261002064214 2101350103217 0000000000000 2101350103217 630391503096321 1050515250501510535 0000000000000 2101350103217 0000000000000 630391503096321 420261002064214 2101350103210 1470721350702114749                                 . Itiseasilyverifiedthemajorproductofasupercolumn vectorwithitstransposeisasupermatrix.

132NeutrosophicSuperMatricesandQuasiSuperMatrices

2 1 0 1 3 1 0 5 7 1 2 0 1



1 2 3 0 1 4 0 5 1 7 3   

At = tt 12AA  =[2101|310|571201]  [12|3014 |051|73].

Havingjustseenthemajorproductofasupercolumnvector withitstransposenowweproceedontoillustratethemajor productofasupercolumnbivectorwithitstransposebya numericalexampleinananalogousway. Example 3.1.30:Let A=A1  A2=

 



                                   

                               beasupercolumnbivector.

ThemajorbiproductofAAt=(A1  A2)(tt 12AA  )which willresultinasymmetricsquaresuperbimatrix.

SuperBimatricesandtheirGeneralization133 = tt 1122 AAAA  = 2 1 0 1 3 1 0 5 7 1 2 0 1                       



[2101|310|571201]  1 2 3 0 1 4 0 5 1 7 3                                   [12|3014|051|73]



              

              

      

 71201

134NeutrosophicSuperMatricesandQuasiSuperMatrices  

222 111

000 111 333



2101310571201        1111 12301405173 2222 3333 0000 12301405173 1111 4444 0000 512530145051573 1111 77 12 33                                  77 301405173 33                     

1210113101571201 000 555 777 111 21013105 222 000 111                           

               

SuperBimatricesandtheirGeneralization135 420262010142402 2101310571201 0000000000000 2101310571201 630393015213603 2101310571201 0000000000000 105051550253551005 147072170354971407 2101310571201 420262010142402 0000000000000 2101310571201                                     12301405173 2460280102146 36903120153219 00000000000 12301405173 4812041602042812 00000000000 51015052002553515 12301405173 71421072803574921 36903120153219                   . Weseethemajorproductofasupercolumnbivectorwith itstransposeasuperbimatrixwhichissymmetric.

Example 3.1.31:Let A=

                                          

136NeutrosophicSuperMatricesandQuasiSuperMatrices

=A1  A2beasupercolumnbivector.At = tt 12AA  =[1010 |213|013751]  [7360–121|003|311];wefind AAt=(A1  A2)(tt 12AA  )=tt 1122 AAAA  = 1 0 1 0 2 1 3 0 1 3 7 5 1

[1010|213|013751] 

17 03 16 00 21 12 31 00 10 33 73 51 11

                    

SuperBimatricesandtheirGeneralization137 7 3 6 0 1 2 1 0 0 3 3 1 1   

[7360–121|003|311]    111 000 1010213013751 111 000 222 1101012131013751 333 000 111 333 10102130 777 555 111                                             13751                         

                 

138NeutrosophicSuperMatricesandQuasiSuperMatrices              777 333 666 7360121003311 000 111 222 111 000 0736012100030311 333 333 1736012110031



   

  311      = 1010213013751 0000000000000 1010213013751 0000000000000 202042602614102 1010213013751 303063903921153 0000000000000 1010213013751 303063903921153 707014721072149357 505010515051535255 1010213013751                                    

111                          

     

    

3.2 Operations on Super bivectors

Inthissectionwedefinethenotionofsuperrowbivectorof typeBandsupercolumnbivectoroftypeB.Wedefinethe basicoperationsofaddition,subtraction,minorandmajor productwheneversuchoperationsarecompatible.Thisconcept ismadeeasytounderstandbynumerousexamples.

Nowweproceedontodefinethenewnotionofsuperrow bivectoroftypeBandsupercolumnbivectoroftypeB.

DEFINITION 3.2.1: Let A = A1  A2 where A1 and A2 are distinct super row vectors of type B then we call A to be a super row bivector of type B (super birow vector of type B).

Wefirstillustratethisbyanexample.

SuperBimatricesandtheirGeneralization139



4921420714700212177 219180363009933 4218240612600181866 0000000000000 7360121003311 146120242006622 7360121003311 0000000000000 0000000000000 219180363009933 219180363009933 7360121003311 73601

21003311                      ItiseasilyseenAAtisasymmetricsuperbimatrix.

140NeutrosophicSuperMatricesandQuasiSuperMatrices



Example 3.2.1:Let A= 3013112345 1100101020 5202030405     

1012331211 0101100223 0512011401 1000102010       = A1  A2.AisasuperrowbivectoroftypeB. Example 3.2.2:Let A= 120010311234 315501020123     1001000 2120111 3000001      =A1  A2;weseeAisasuperrowbivectoroftypeB. Example 3.2.3: LetA=A1  A2= 00000000 00000000 00000000 00000000        000000000 000000000 000000000      ;

AisasuperrowbivectoroftypeB.WeseeAisazerosuper rowbivectoroftypeB.

Example 3.2.4: LetA=A1

A2=

beasuperrowbivectoroftypeB.

WecallthisasasuperunitrowbivectoroftypeBorsuper rowunitbivectororbisuperunitvectoroftypeB.

Wenowproceedontodefinethenotionofsupercolumn bivectoroftypeB.

DEFINITION 3.2.2: Let A = A1  A2 where A1 and A2 are distinct super column vectors of type B. We call A to be a super column bivector of type B.

Weillustratethisbythefollowingexamples.

SuperBimatricesandtheirGeneralization141



11111111111 11111111111 11111111111 11111111111 11111111111          111111111 111111111 111111111          

142NeutrosophicSuperMatricesandQuasiSuperMatrices

A=A1  A2= 301 30145458 21037767 12345841 54321110 01010202 10101010 12301123 40512456 37200789 012                                                  ;

Wegiveyetanotherexamplebeforeweproceedtogive

A=A1  A2= 31 123021 111012 000113 234530 678910 310211 201101 51                                         ;

Example 3.2.5: Let

AisasupercolumnbivectoroftypeB.

examplesofthenotionofsuperzerocolumnbivectorandsuper columnunitbivectoroftypeB. Example 3.2.6: Let

AisasupercolumnbivectoroftypeB.

Example 3.2.7: Let

A=A1  A2=

                                                        

00 00 00000 00000 00000 00000 00000 00000 00000 00000 00000 00 00

;

AisdefinedtobeasuperzerocolumnbivectoroftypeB.

Example 3.2.8:Let A=A1

SuperBimatricesandtheirGeneralization143

000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 

                              

 A2=



; AisnotasuperzerocolumnbivectoroftypeBasA1=A2.

Example

AisasuperzerocolumnbivectoroftypeB.

Example

AisasuperunitcolumnbivectoroftypeB.

NowasincaseofsuperrowbivectorsoftypeBwecan definethetransposeofasuperrowbivectoroftypeB.Itiseasy toseethatthetransposeofasuperrowbivectoroftypeBisa supercolumnbivectoroftypeBandviceversa.

144NeutrosophicSuperMatricesandQuasiSuperMatrices

3.2.9:LetA=A1  A2= 0000 000000 0000 000000 0000 000000 0000 000000 0000 000000 0000 000000 0000 000000 0000 000000 0000 000000 0000                                              ;

3.2.10: LetA= 1111 1111111 1111111 1111111 1111111 1111111 1111111 1111111 1111111 1111                                               ;

SuperBimatricesandtheirGeneralization145 Weshallillustratethissituationbysomesimpleexamples. Example 3.2.11:LetA=A1  A2= 3120301013 1011410100 0120512345       3110345 1111234 0100567 1126011             beasuperrowbivectoroftypeB. NowtransposeofAdenotes byAt=(A1  A2)t = tt 12AA  310 101 3101 212 1111 010 1102 345 0106 011 3250 102 4361 013 5471 104 305                                        .

ItiseasilyseenthatAtisasupercolumnbivectoroftypeB.

146NeutrosophicSuperMatricesandQuasiSuperMatrices Example 3.2.12:Let A= 21 35 013 67 111 01 021 12 456 34 701 56 301 01 084 23 011 45 110 67 101 89 111 12 03                                                 



      beasupercolumnbivectoroftypeB. ThetransposeofAdenotedbyAt = tt 12 01047300111 AA11250081101 31161141011        23601350246810 15712461357923    . ClearlyAtisasuperrowbivectoroftypeB. NowasincaseoftypeAsuperrow(column)bivectorswe candefineidenticalinstructuresuperrow(column)bivectorsof typeB.



 

 

DEFINITION 3.2.3: Let A = A1  A2 where A1 and A2 are distinct i.e., A = A1  A2 is a super row bivector of type B. Let B = B1  B2 be a super row bivector of type B.

We say A and B similar or “identical in structure” if the length of Ai and Bi are the same and the number of rows in both Ai and Bi are equal and the partitions in Ai and Bi are identical for i=1,2.

By the term partitions are identical we mean if in Ai the first partition is say between the r and (r+1)th column then in Bi also the first partition will be only between the r and (r+1)th column and the number of partition carried out in Ai will be the same as that carried out in Bi, i=1,2.

Wewillillustratethisbysomesimpleexamples.

Example 3.1.13:Let

SuperBimatricesandtheirGeneralization147

A=A1  A2= 301237120132 110101234015 000511000178            23231254 31002310 15000000 01711237             and B= 110012003421 010000560001 000170000810      

148NeutrosophicSuperMatricesandQuasiSuperMatrices



 betwosuperrowvectorsoftypeB.WeseeAandBare

Example 3.2.14:Let A=A1  A2= 200113378 116215123        57838800513 91241168157 03730074278      and B=B1  B2= 135678438 249012259        10279893155 34501203577 67862007119      betwosuperrowbivectorsoftypeB.WeseeAandB

10111001 21001110 31110110 50001111  

 

“identicalinstructure”.ForthebilengthofAandBis(12,8). AlsothepartitionsareidenticalinBiandAi,i=1,2.The numberofrowsinAiequaltothenumberofrowsinBi,i=1,2.

“identicalinstructure”.

SuperBimatricesandtheirGeneralization149 Example 3.2.15:Let A=(A1  A2)= 3123781711 5791015822 0127156933 1101234044              3417017084 0700011256 1008623708      andB=B1  B2= 1111013213 2020826127 3021100136            213710438 487623675    betwosuperrowbivectorsoftypeB.WeseeAandBdoesnot haveidenticalstructure. Wedefinenowadditionoftwosuperrowbivectorsoftype Bofidenticalstructure. DEFINITION 3.2.4: Let A = A1  A2 =           111111 11 1r1r1 1s 1t 1m 111111 n1 nrnr1 ns nt nm aaaaaa aaaaaa          11111 11 1u 1r 1s 1w 11111 p1 pu pr ps pw bbbbb bbbbb

is defined to be the sum of the two super row bivectors A and B of type B. Clearly A+B is again a super row bivector of type B with identical structure with A and B. Nowinviewofthiswehavethefollowinginteresting theorem.

THEOREM 3.2.1: Let X = {A = A1  A2 be the collection of all super row bivector of type B which are identical in structure with A with entries from the field Q or R or C or Zp (p a prime) or from the rings Z or Zn, n not a prime or C(Zp) or Z(g1) or

150NeutrosophicSuperMatricesandQuasiSuperMatrices and if B = B1  B2 = 222222 11 1r1r1 1s 1t 1m 222222 n1 nrnr1 ns nt nm aaaaaa aaaaaa            22222 11 1u 1r 1s 1w 22222 p1 pu pr ps pw bbbbb bbbbb         are



            121212121212 11111r1r1r11r11s1s1t1t1t1t 121212121212 n1n1nrnrnr1nr1nsnsntntntnt

 

 

two super row bivectors of type B. We see A and B are identical in structure. The sum of A and B denoted by A+B = (A1

A2) + (B1

B2) = A1 + B1  A2 + B2

aaaaaaaaaaaa aaaaaaaaaaaa  1212121212 1111 1u1u 1r11r11s1s1w1w 1212121212 p1p1pupu pr1pr1psns ptpt bbbbbbbbbb bbbbbbbbbb

      

SuperBimatricesandtheirGeneralization151 Z(g2) or Z(g3) (g1, g2 and g3 are new elements such that 2 g1 = 0, 2 2 g = g2 and 2 3 g = –g3 and Z can be replaced by Q or R or C or C(Zn) or Zn}. Then X is a group under addition. Example 3.2.16:LetX=X1  X2= 321010013401 152601231001 063061000100       31414130021011 63212011102011    and Y= 310010110401 012010201001 013101010111       30103010121011 21110201011101    =Y1  Y2betwoidenticalstructurewiththesuperrow bivectoroftypeB.Tofindthesum X+Y=(X1  X2)+(Y1  Y2)=(X1+Y1)  (X2+Y2) = 031000123000 144611432000 076162010011       01517140140022 84102212112112    . WeseeX+Yisagainasuperrowbivectorwithidentical structureasthatofXandY.ThesubtractionofXwithY

Nowinasimilarwaywecandefinethenotionwhenare twosupercolumnbivectorsoftypeBare“identicalin structure”.LetA=A1  A2andB=B1  B2betwosuper columnbivectorsoftypeB.WesayAandBaresupercolumn bivectorsoftypeBareidenticalinstructureifthenumberof naturalrowsinAiisequaltoBi,i=1,2andthenumberof columnsinAiequaltoBiandthepartitionofAiandBiare identical.Wecanasincaseofidenticalstructuresuperrow bivectorsoftypeB,wecanincaseofidenticalstructureof supercolumnbivectorsoftypeB,wecandefineadditionand subtractionofsupercolumnbivectorsoftypeB.

152NeutrosophicSuperMatricesandQuasiSuperMatrices X–Y=(X1–Y1)  (X2–Y2) = 611020103802 160611030002 050160010211       61311120102000 42322210112110    .

Example 3.2.17: LetA= 3100328 0011060 4021110 3200101 1012641 1234001 6702702 0231080 1011103 4256462 0120675 0183320                   and

X–YisagainasuperrowbivectoroftypeB.

SuperBimatricesandtheirGeneralization153 B= 0100012 0010020 1010110 0101010 1102011 0036110 1124102 0013063 1000101 2011215 8001010 0114025                                        betwosupercolumnbivectoroftypeB. ThesumofA+B=(A1+B1)  (A2+B2) 32003310 0021080

5011220 3301111 1114652 12610111 7826804 02440143 2011204 6267677 8121665 0297345                     isagainasuperbicolumnbivectoroftypeB. Inviewofthiswehavethefollowinginterestingtheorem.

154NeutrosophicSuperMatricesandQuasiSuperMatrices THEOREM 3.2.2: Let V = {X = X1  X2 / collection of all super column bivector of type B with entries from a field or ring which are identical in structure}. Then V is a group under addition. Howevertodefinemultiplicationwehavetodefineeither minorproductormajorproduct. Weproceedontogivetheminorproductoftwosuper bivectorsoftypeBbynumericalillustrations. Example 3.2.18:Let X=(X1  X2)= 15325701 06211011 27100101            12010112340 01102101001 01010500112 10103011001       andY= 1201 0115 310 0010 100 2100 001 1001 002 0110 101 1001 221 0101 453 1010 012 0201 0120                                                 =Y1  Y2

SuperBimatricesandtheirGeneralization155 betwosuperbivectors. TheminorproductofXwithYisgivenby XY=(X1  X2)(Y1  Y2) =X1Y1  X2Y2. = 310 100 001 15325701 002 06211011 101 27100101 221 453 012                                 1201 0115 0010 2100 12010112340 1001 01102101001 0110 01010500112 1001 10103011001 0101 1010 0201 0120                                         =  1533102 0621001002 2710010       

156NeutrosophicSuperMatricesandQuasiSuperMatrices 10112011201 5701 22101100115 1011 45301010010 0101 01210102106             1001 0112340 0101 21100101001 1010 05011000112 0201 3011001 0120                           = 813004191514 602002566 1321000233       0110352141037 211201250221 055022151451 300312111222        = 71668 271621 2458 11610 311116 1554 5435                   . Itispertinenttomentionthattheentriesofthesesuper matricescanbereplacedbyneutrosophicrings,specialduallike numberneutrosophicrings,specialquasidualnumber neutrosophicringsandmixedspecialdualnumberneutrosophic rings.

QUASI SUPER MATRICES

Asquareorarectangularmatrixiscalledasaquasisuper matrix.Itcannotbeobtainedbypartitionofamatrix.Allsuper matricesarequasisupermatrixbutquasimatrixingeneralis notasupermatrix.Thustheelementsofaquasimatrixare submatricesfollowingsomeorder.

Chapter Four

DEFINITION 4.1: Take a m × n matrix A with entries from real or complex numbers. A =                   1 2 tttm 1112 1r 2122 2r s1s2 sr AAA AAA AAA where Aij are submatrices of A with 1  j  r1 , r2, …, rm and 1  i  s1, s2, …, st

Clearly we do not demand the size of Aji to be same but the only condition to be satisfied is that if A is a m × n matrix then the number of the rows in each column of the matrix adds up to m and the number of the columns in each of rows added up to n. We define A to be a quasi super matrix.

Further we do not demand r1 = r2 and ri = rj in general also. Also si  sj in general. Wegiveexamplesofoneortwoquasisupermatrixbefore theproceedontodefinesomeoftheirbasicproperties.

158NeutrosophicSuperMatricesandQuasiSuperMatrices

A= 3212 0150 9832 1414 2530 3151           = 1112 2122 AA AA       where A11= 321 015       ,A12= 2 0       , A21= 98 14 25 31             andA22= 32 14 30 51             . WeseethesumoftherowsofA11andA21addsupto4andsum oftherowsofA12andA22addsupto6whereassomeof columnsaddsupto4i.e.,thesumofthecolumnsofA11andA

21andA22isalso4.

Example 4.1: LetAbea6×4quasisupermatrixgivenby

12 is4andthatofA

QuasiSuperMatrices159

 

= 111213 212223 31323433 AAA AAA AAAA           ClearlythenumberofcolumnsaddedupinA11A12andA13 to4+2+2=8. ThesumofthecolumnsaddedupinA21A22andA23to4+ 3+1=8.ThesumofthecolumnsofA31A32A34andA33to3+ 1+3+1=8. WeseebasicallyAisa8×8matrixbutbythedivisionitis madeintoaquasisupermatrixofthegivenformhaving10 numberofsubmatrices.A11,A12,A13,A21,A22,A23,A31,A32, A33andA34. Nowany8×8matrixcanformadivisioninthefollowing formgivenintheexample4.3.

Nextwegivesomemoreexamples. Example 4.2: LetAbe9×8quasisupermatrixgivenby A= 20311230 05212111 31201 310 12110 031 33002 202 51111 303 01220 110 11014 101

                                      

HerethesumoftherowsofthesubmatricesA11andA21is 3+5=8andsumofthecolumnsofthesubmatricesA21and A22is5+3=8.

Wemakethefollowingobservationweseeallthefoursub matricesaresquarematricesbasedontheseobservationswe makethefollowingdefinition.

DEFINITION 4.2: Let A be the m × m square matrix. If all the sub matrices of A are also square matrices then we call A to a quasi super square matrix.

Thematrixgiveninexample4.3isasquarequasisuper matrix.Wegiveyetanotherexamplebeforewedefinesome moreresults.

Example 4.4: LetAbea8×8matrixwiththefollowing representation.

160NeutrosophicSuperMatricesandQuasiSuperMatrices





     

  



      

    



Example 4.3: LetAbeany8×8matrix. A= 52102 03111021 15200110 00152001 11220 19312 01010120 10101315 22025115 31020

 





  

   





            = 1112 2122 AA AA       .

SumoftherowsA11andA21=8, SumoftherowsA11andA23=8, SumoftherowsA11andA24=8, SumoftherowsA12,A22,A23,A25is8, SumofthecolumnsofA11andA12is8, SumofthecolumnsofA11andA22is8, SumofthecolumnsofA11andA23is8, SumofthecolumnsofA21,A31,A24andA25is8.

QuasiSuperMatrices161

        

A= 03152102 15211001 00100110 79312101 01010120 10101320 22025315 31020115                                       = 12 1122 23 21312425 A AA A AAAA   

Example 4.5: LetAbea5×6matrixwhichisgiveninthe following. A= 301561 110325 212102 210111 021011                           

162NeutrosophicSuperMatricesandQuasiSuperMatrices =

     

         

Now A11= 0312 1105 5121 0101             , A12= 01 21       , A21= 11 02    ,A°=A°= 00 00       and

1112 212223

AAA

whereA11,A12,A21,A22andA23aresubmatrices. ClearlysumofrowsofA11andA21is5,sumofrowsofA12 andA22isfiveandthatofA12andA23isfive. SumofthecolumnsA11andA12issix.Sumofthecolumns ofA21A22andA23issix. Example 4.6: LetAbea6×6quasisupermatrixwith A= 1112 00 2122

AA AA

, whereA11isa4×4squarematrixandA12isa2×2square matrix.A21isa2×2squarematrixandA22isa4×4square matrix.

QuasiSuperMatrices163 A22= 6012 1002 1100 0101             . A= 031201 110521 512100 010100 006012 001002 111100 020101                           .

A=  010 20 100 10 111 11 101 01                                 = 1112 2122 AA AA       .

Weseeallthesixsubmatricesaresquarematrices. Example 4.7: Abea4×5matrixwhichisaquasisupermatrix givenby

Similarlyweproceedontodefinethenotionofquasisuper matrixwhichisrectangular. DEFINITION 4.3: Let A be a quasi super matrix.

Aij’s are rectangular matrices, with 1 < i < r1, r2, …, rm and 1 < j < s1, s2,…, st, we define A to be a quasi super rectangular matrix.

Wegiveanexampleofamatrixofthistype.

                   AA AA      

Aisarectangularquasisupermatrix.

Example 4.9: LetAbea6×4matrix.

20121100 11051011 11050111 10111010 11110001 20122022 = 1112 2122

164NeutrosophicSuperMatricesandQuasiSuperMatrices

A = 1 2 12tm 1112 1r 2122 1r s1s2 sr AAA AAA AAA                  

Example 4.8: LetAbea6×8matrixquasisupermatrix. A=

A= 2101 1120 1201 1025 3102 1110                                            

whichisaquasisupermatrix.SumofthecolumnsofA11and A12isfour. NumberofcolumnsinA21isfour.SumofrowsofA11and A21issix.SumofrowsofA12andA21issix.Thequasisuper matrixdescribedinexample4.9isnotarectangularquasisuper matrixforitcontainsa3×3squarematrixinit. Weproceedontodefinethesetypeofmatrices,.

DEFINITION 4.4: Let A be a m × n rectangular matrix. A be a quasi super matrix. A =

1112

2122

AAA AAA AAA If some of the Aij’s are square submatrices and some of them are rectangular submatrices, then we call A to be mixed quasi super matrix.

Thuswehaveseen3typesofquasisupermatrices. Wehavetogivesomemoreexamplesforwecanhavea squarem×mmatrixA:yetAcanbeamixedquasisuper matrixorarectangularquasimatrix.Likewiseasquarem×m matrixcanbeamixedquasisupermatrix.

Considera6×6quasisupermatrixwhere

QuasiSuperMatrices165

= 1112 21 AA A      

            

    

1 2 12tm

1r s1s2 sr

Example 4.10:

A=1112 2122 AA AA      

Aisa6×6matrixbutAisnotarectangularquasisuper matrixorasquarequasisupermatrixitisonlyamixedquasi supermatrix.

Thusthisexampleclearlyshowsaquasisupermatrixwhich isa6×6squarematrixcanbeaquasisupermixedsquare matrix.

Nextweproceedontogiveanexampleofasquarematrix whichisonlyarectangularquasisupermatrix.

Example 4.11: LetAbea5×5quasisupersquarematrix.

166NeutrosophicSuperMatricesandQuasiSuperMatrices where A11=             21 12 6025 4102 =A12 A21= 312 101 010 210       andA22= 101 010 111 022             i.e.,A= 201421 520612 312101 101010 010111 210022                     .

QuasiSuperMatrices167 A= 01234 12340 23401 34012 40123                 = 1112 2122 51 AA AA A           where A11= 012 123       ,A12= 34 40 01           A21= 234 340       ,A22=[12]and A31=[40123]. Aisaquasisuperrectangularmatrix. Example 4.12: LetAbea6×4quasisuperrectangularmatrix. A= 0123 1102 2010 1671 5208 3192                     = 1112 21 AA A       .

ClearlyAisasquarequasisupermatrixwhichisnotasquare matrix.WeseeallthethreesubmatricesofAaresquare matrices.

Nowhavingseenthreetypesofquasisupermatrices,we nowproceedontodefinesomesortofoperationonthen.First givenanyn×mmatrixwhichwearenotinapositiontogivea partitionasincaseofsupermatrices.

Sowedefineincaseofn×mmatricesanewtypeof relationcalledquasidivision.

Aquasidivisionontherectangulararrayofnumbersisa cellformationsuchthatthecellsareeithersquare,rectangular orroworcolumnorsingletons.Clearlyorisnotusedinthe mutuallyexclusivesense.

Wejustillustratethisbyaverysimpleexample.

Example 4.13: LetAbea7×6quasisupermatrix.

Weseethematrixorthis7×6arrayofnumberswhich havebeendividedin10cells.Eachcellisofavaryingsize.For weseefirstcellisa2×2squarematrix.

168NeutrosophicSuperMatricesandQuasiSuperMatrices

A= 078941 101010 111101 801210 011101 211400 100111                       .

07 10       =Icell=A11

Thesecondcellisarectangular2×3matrix.

QuasiSuperMatrices169

894 101      

1 0   

1 8 0           =A

istheIIcell-A12

istheIIIcell-A13isa2×1columnvectorormatrix. Thefourthcellisagainacolumnvectorgivenby

2 1      

012 111 114 001             =A32

10 01 00           =A33.

Thefifthcellis[1],asingletonA22.Thesixthcellis[–1101] isarowvectorA23. Theseventhcellisacolumnvector

=A31.Theeighthcellis a4×3rectangularmatrix.

Theninthcellisa3×2rectangularmatrix.

The10thcellfinditsplaceasA44arowvector[11].The arrangementofwritingornotationalorderiswritteninthisform

WeseethesumofthecolumnsofA11,A12andA13is6.Sumof thecolumnsofA21,A22andA23issix.

SumofthecolumnsofA31,A32andA33issix.(1+3+2=6).Also sumofthecolumnsofA31,A32andA44issix.

Nowhavingseenasexamplewedefinethenotionofcell partition.

DEFINITION 4.5: A cell partition of m × n rectangular array of numbers A is a division of the m × n array of numbers using cells. The cells can only be singletons or row vectors or column vectors or square matrix and (or) rectangular matrix. i.e., each cell can be called as a sub matrix of A.

Forinstancethisprocessislikefindingsubsetsofaset.In thatcasewehavelotsofchoicesandweknowgivenNnumber ofelementsintheset;thenumberofsubsetsis2N.Butdividing then×marrayofnumbersofthen×mmatrixisnotidentical forthecelldivisionisnotarbitraryeachcellshouldhaveaform andthedivisioncanbemanyforinstancewewillfirstillustrate inhowmanywaysa2×2matrixcanbedividedusingthe methodofcelldivision,

Weenumeratethenumberofcelldivisionleadingthequasi supermatrix.

170NeutrosophicSuperMatricesandQuasiSuperMatrices A=

           

111213 212223 313233 44 AAA AAA AAA A

     

Example 4.14: LetA= 01 23

      = 11 12 21 A

     

A A

QuasiSuperMatrices171 A2= 01 23       = 1112 22 AA A       , A3= 01 23         andA4= 01 23         Thereareonly4quasisuperma

Howeverwehaveseveralpa

B= 01 23     isapartition,

C= 01 23    isasupermatrixdifferentfromB. D= 01 23     isasupermatrixdifferentfrombothBandC. Thususing2 

tricesconstructedusinga 2×2matrix.

rtitionsleadingtoasuper matrix.

Bisasupermatrix.

2matriceswehaveonly3supermatrices. Butusingcelldivisionleadingtoquasisupermatrixwehave4 quasisupermatrices.Thuswehavealtogether7quasisuper matricesifwemaketheruleallsupermatricesarequasisuper matrices.Thatistheclassofsupermatricesiscontainedinthe classofquasisupermatrices.

172NeutrosophicSuperMatricesandQuasiSuperMatrices

101 111          

012 101 111           = 1112 22 AA A      

111           = 1112 22 AA

     

          = 12 1122 13 A AA

          A4= 012 101 111      = 13 1112 23 A AA A       A5= 012 101 111      = 13 1112 23 A AA A       A6= 012 101 111      = 111213 23 33 AAA A A          

Example 4.15: Let A= 012

bea3×3matrix. ThenumberofquasisupermatricesfromAare A1=

A2= 012 101

A3= 012 101 111

QuasiSuperMatrices173

7= 012 101 111           = 111213 22 AAA A      

A8= 012 101 111           = 111213 22 AAA A       A9= 012 101 111           = 111213 22 32 AAA A A          

A10= 012 101 111           = 111213 2223 AAA AA       A11= 012 101 111           = 111213 2223 AAA AA       A12= 012 101 111           = 111213 2223 33 AAA AA A           A13= 012 101 111           = 111213 2223 AAA AA      

174NeutrosophicSuperMatricesandQuasiSuperMatrices

012 101 111      = 111213 2223 AAA AA       A

012 101 111      = 111213 2223 33 AAA AA A           A16= 012 101 111      = 111213 2223 32 AAA AA A           A

012 101 111      = 111213 2223 32 AAA AA A           A

012 101 111      = 111213 2223 33 AAA AA A           A19= 012 101 111      = 111213 212223 3233 AAA AAA AA           andsoon. Thusweseeevenincaseofa3×3matrixthenumberof quasisupermatrixisverylarge.Ifwemaketheinclusionof supermatrixintoit.Itwillstillbelarger. Thusatthisjunctureweproposethefollowingproblem. Problem: SupposePisanym×msquarematrix.i.e.,asquare arrayofm×mnumbers.

A14=

15=

17=

18=

QuasiSuperMatrices175 1.FindthenumberofsupermatrixconstructedusingP. 2.Findthenumberofquasisupermatrixconstructed usingP. Nowwejusttrytoworkwitha2×4matrix. Example 4.16: LetM=       0101 3210 bea2×4matrix. Wejustindicatesomeofthequasisupermatrixconstructed usingM. M1= 0123 1010       = 12 11 22 A A A       M2= 0123 1010       = 13 1112 23 A AA A       M3= 0123 1010    = 11121314 23 AAAA A       M4= 0123 1010    = 111213 2324 AAA AA       M4= 0123 1010       = 111213 2324 AAA AA       M5= 0123 1010    = 11121314 23 AAAA A       M6= 0123 1010       = 1112 22 AA A      

andsoon.Thusweseeevenincaseofthesimple2×4matrix, wecanhaveseveralquasisupermatrices.Wehaveshownonly nineofthem.

Nowweproposethefollowingproblem.

Problem

1.Findthenumberofquasisupermatricesthatcanbe constructedusingm×nmatrix.

2.Findthenumberofsupermatricesthatcanbe constructedusingjustam×nrectangularmatrix.

Nowhavingdefinedthenotionofcellspartitionofa matrixwhichhasleadtothedefinitionofquasisuper matricesweproceedonthestudythefurtherproperties ofquasisupermatriceslikematrixadditionand multiplicationandseehowbestthoseconceptscanbe definedonthem.

Example 4.17: LetAbea9×4rectangularmatrix.

176NeutrosophicSuperMatricesandQuasiSuperMatrices

 

   



M7= 0123 1010    = 111213 22 AAA A    

M8= 0123 1010    = 1112 2223 AA AA  

M9= 0123 1010    = 111213 21 AAA A     

QuasiSuperMatrices177 A= 0125 1101 0111 1010 0101 1100 1001 2135 1234                             = 1112 2122 32 414243 AA AA A AAA             . WeseeAhas8cells.WecanhaveforthesameAwecanjust

A= 0125 1101 0111 1010 0101 1100 1001 2135 1234                             = 1112 2122 AA AA       .

triceshowtoadd?Wefirst

haveonly4cells.

Supposewehavetwoquasisuperma

proceedontodefineadditionofasuperquasimatrixAwith itself.

178NeutrosophicSuperMatricesandQuasiSuperMatrices Example 4.18: LetAbea5×7quasimatrixwithawell definedcellpartitiondefinedonit. A= 0123456 1101101 0110110 5782143 8170258         = 111213 22 313233 AAA A AAA           . A+A=2A= 024681012 2202202 0220220 1014164286 16214041016                 = 111213 22 313233 2A2A2A 2A 2A2A2A      =2A. Itisleftasanexerciseforthereadertoverifythatforthequasi supermatrixAwehaveA–A=[0].Inthiscase A–A= 0000000 0000000 0000000 0000000 0000000                 .

JustaquasisupermatrixAunderthesamecelldivisioncan beaddedanynumberoftimesandthiswillnotchangethe natureofaquasisupermatrixcelldivision.

ThuswecansayifAisaquasisupermatrixthenA+…+ A:ntimesisthesameasnA.

IfAhasA11,A12,…,Arttobethecollectionofallits submatricesthennAwillhavenA11,nA12,…,nArttobethe collectionofallitssubmatrices.

Note: Wecanalwaystakeanym×nzeromatrixandmakethe cellpartitionordivisioninadesiredformsothatA+(0)=(0)+ A=A.Forthiswefirstdefineasimplenotioncalledcell divisionfunction.

Supposewehavetwom×nmatricesAandB.Weknow thecelldivisionofAhowtogetthesamecelldivisionofBso thatthecelldivisionsofbothAandBarethesameandbothof themarequasisupermatricesofsametypeorsimilar.

DEFINITION 4.6: Let A be a m × n quasi super matrix. B any m × n matrix. To make B also a quasi super matrix similar to A; define a cell function Fc from A to B as follows.

Fc(A11) = B11 formed from B by taking the same number of rows and columns from B. So that A11 and B11 have the same number of rows and columns, not only that the placing of B11 in B is identical with that of A11 in A.

The same procedure is carried out for every Aij in A.

Thus the cell function Fc converts a usual m × n matrix B into a desired quasi matrix A.

BeforeweproceedontodefinethepropertiesofFcwe showthisbysomeillustrations.

QuasiSuperMatrices179

180NeutrosophicSuperMatricesandQuasiSuperMatrices

          = 1112 21222324 AA

     

35111           DefinethecellfunctionFcfromAtoBasfollows. Fc(A11)= 00.3 14       =B11, Fc(A12)=[40.12]=B12, Fc(A21)=[351]=B21, Fc(A22)=[0]=B22, Fc(A23)= 2 1       =B23

Example 4.19: LetAbeagiven3×5quasisupermatrix.Bbe anymatrix.TofindasimilarcelldivisiononBidenticalwithA. Given A= 03520 14412 25304

AAAA

isaquasisupermatrixwithA11,A12,A21,A22,A23andA24as submatrices. Given B= 00.340.12 14020

WeenumeratesomeofthepropertiesofFc.LetFcbeacell divisionfunctionfromAtoBwhereAisaquasisupermatrix. Fc(A)=A i.e.,AisequivalenttoAunderFc;i.e.,Fcisreflexive. IfFc(A)=B,i.e., AisequivalenttoBunderFcthenBisequivalenttoA underFc. i.e.,ifFc(A)=BthenFc(B)=A. (Fc(Fc(A))=AforallA). IfAisanyquasisupermatrixandBandCanytwom×n matrices.IfFcisthecelldivisionfunctionsuchthatFc(A)  B sothatBbecomesaquasisupermatrix.SupposeFc(B)  Cso thatCbecomesaquasisupermatrix,thenweseeA,BandC andquasisupermatriceswith A~BandB~CimpliesA~C. ThusFcisanequivalencerelationontheclassofallm×n matricesunderaspecialoraspecifiedcelldivisionfunction.

QuasiSuperMatrices181

    

  

andFc(A24)= 0 1       =B24. B= 00.340.12 14020 35111     

= 1112 21222324 BB BBBB   

Ifwetakethecollectionofallcellspartitionsofam×n matrixanddenoteitby cF Cthen cF Cisdividedintodisjoint classesbythisrelationi.e., cF C={Fc|Fc;A  B;AandB m×nmatrices,AisaquasisupermatrixandBanym×n matrix}

182NeutrosophicSuperMatricesandQuasiSuperMatrices

T= ab cd     

    

       2

      3 c

        4 c

       

Weillustratethisbyanexample. Example 4.20: Let

a,b,c,d  Z2={0,1}}. TofindthepartitionofTunderthecelldivisionfunction. Firstwefindthenumberofelementsin cF C. 0 c F= ab cd 

1 c F= ab cd 

c F= ab cd

F= ab cd

QuasiSuperMatrices183 5 c F= ab cd         6 c F= ab cd       7 c F= ab cd       Wesee cF C={0 c F,1 c F,2 c F,3 c F,4 c F,5 c F,6 c F,and7 c F, i.e.,| cF C|=8. NowwehavetofindthenumberofmatricesinT. T= 0011100100001110 ,,,,,,,, 0011000010010010         0100100111110110 ,,,,,,,. 0111011010011111         Thuseachandeveryclassin cF Cwillhaveallthe16 elementsofTwithnooverlap. If cF Cistheclasstheneveryclasscontainsexactlythesame numberofelementsinthesetofmatricesonwhichthequasi supermatricesnotionistobedefined.

divisionleadstotheconceptofquasisupermatrix.

Nowaspartitionyieldstoasupermatrixweseecell

Nowtwom×nquasisupermatrixcanbeaddedifandonly iftheyhavethesamecelldivisiondefinedanthem,otherwise

184NeutrosophicSuperMatricesandQuasiSuperMatrices wesayadditionisnotcompatibleforthecelldivisionisnot compatible. Example 4.21: Let A= 30125 71011 60010 30100 12511 11621                     = 1112 212213 31 AA AAA A           ; whereA11,A12,A21,A22,A13andA31aresubmatricesofA. B= 11002 50110 62102 11011 20101 30220           = 1112 2122 31 BB BB B           . ClearlytheadditionofAandBisnotcompatiableforwe seethesubmatricesB11,B12,B21,B22,andB31aredifferentfrom A11,A12,A21,A22,A13andA31. ThusweseeadditionofAwithBisimpossiblethoughboth AandBareofsameorderviz.,6×5. Thusisaquasisupermatrixadditioniscompatiableifand onlyif(1)AandBarequasisupermatrixofsameorderand

QuasiSuperMatrices185 (2)celldivisiononAandBarethesamethatiswehaveacell divisionfunctionFc:A  BsuchthatFc(A)=Btrue. Nowweillustratethissituationbyasimpleexample. Example 4.22: Let A= 01234 56789 06284 51739 11012 02101                     = 1112 212223 31 AA AAA A           whereA11,A12,A22,A23andA31aresubmatrixofA. B= 10101 01010 11111 11001 10011 01001                     = 1112 2223 31 BB BB B           WeseethetotalnumberofcolumnsinB11andB12is5;in B22andB23is5inB31andB23is5.

186NeutrosophicSuperMatricesandQuasiSuperMatrices NowtheadditionofAandBiscompatible. A+B= 01234 56789 06284 51739 11012 02101           + 10101 01010 11111 11001 10011 01001                     = 0110213041 5061708190 0161218141 5111703091 1110001121 0021100011                 = 11335 57799 17395 627310 21023 03102                     . Itisimportantandinterestingtonotethatincaseofquasi supermatricesevenadditionisnotalwayscompatibleevenif thematricesareofsameorder. Nowweproceedontodefinemultiplicationincaseofquasi supermatrices. Whateverbethesituationweneedtohavecompatabilityof orderwhilemultiplying.

QuasiSuperMatrices187 Example 4.23: Let A= 54 01221 51131 10100 51011                and B= 5x4 1011 0102 1010 2101 0011                 . beanytwoquasisupermatrices. AB= 012211011 511310102 101001010 510112101 0011                         Ifthepartitionsorcelldivisionsareremovedimaginarily andthemultiplicationtakesplacewegeta4×4matrix. AB= 4335 124711 2021 7269              

Thequasisupermatrixproductassumesanapproximatecell divisionwhichisnotuniqueforitmaynotbepossibletogeta celldivisionexactlyastheveryorderofitischanged.

Wecallitastheresultantquasisupermatrix.Whatismore importantinthissituationisthatwecanfindoneormoretype ofcelldivisionontheproduct.Thisisnotaproblemforthe veryconceptofquasisupermatrixwasfoundtoapplyinfuzzy models.

Sotoovercomeallthesehurdleswemakeamentionofthe following.Asincaseofsupermatriceswefirstapplythe productofaquasisupermatrixanditstransposewemayrecall incaseofsupermatrixwehadtheproductofasupermatrix withitstransposegaveanicesymmetricsupermatrix.

WeobservethatinthecaseofquasisupermatrixA,the transposeATofAissuchthatingeneraltheproductisnot defined.Thuswecandefinetheproductonlywhenitisdefined.

Thisisnotthecaseincaseofsupermatrix,forinsuper matrixwecanalwaysdefinesuchproductsbutincaseofquasi supermatrixtheproductmaybedefinedornot.Onlyifdefined itcanbecalculated.

Thisquasisupermatrixwhenitsentriesarefromthefuzzy interval[0,1]wedefineittobeafuzzyquasisupermatrixor quasifuzzysupermatrix.

Allthesematricesfindtheirapplicationswhentheproblem underinvestigationusesseverali.e.,oneormorefuzzymodels simultaneously.

Wejustillustratebyexamplessomefuzzyquasisuper matrices.

Example 4.24: LetFsdenotethefuzzyquasisupermatrixgiven as

188NeutrosophicSuperMatricesandQuasiSuperMatrices

QuasiSuperMatrices189

Fs= 1 3 2 A A A       = 00.10.40.50.30.7 10.1100.80.5 0.70.20.50.610.2 0.80.50.60.50.71 0.10.80.6100.8 0.80.50.10.910.2 0.30.70.50.30.51                       , whereA1isa4×6fuzzyrectangularmatrixandA2andA3are 3×3fuzzysquaresupermatrices. Example 4.25: LetPsdenoteaquasifuzzysupermatrixgiven by Ps= 0.110.20.10.90.70.20 0.90.1100.80.60.40.6 0.70.50.60.80.70.10.50.1 0100.50.40.70.40.3 0.310.60.10.5000 0.700.20.40.60.700 0.81101011 0.900.70.80.90.60.30.2 0.50.20.60.40.50.10.30.1 0.60.30.40.30.20.70.10                      ; where Ps= 12 4 3 5 AA A A A           ;

whereA1andA2isa4×4fuzzymatrix.A3isa6×6square fuzzymatrixandA4andA5are3×2rectangularfuzzy matrices. Thuswecanhaveanynumberofquasifuzzysuper matrices.Incaseofevenquasifuzzysupermatriceswecannot alwaysdefinethenotionofadditionormultiplication.Only whenspecialtypeofcompatibilityexistswecanproceedonto definesumorproductorboth.

Example 4.26: LetFsandPsbetwoquasifuzzysupermatrices givenby Fs=

0.30.10.20.70.200.1 10.20.5000.51 00.310.10.10.20.3 00.30.80.410.10.7 10.70.310.10.71 0.50.80.900.30.20.5

AA AAA         and Ps=

          = 12 345

00.7100.610.7 0.300.810.50.31 0.10.90010.20 0.900.70.50.10.11 0.20.30.50.30.510.7 0.40.10.61000.5

          = 12 345

BB BBB        

190NeutrosophicSuperMatricesandQuasiSuperMatrices

Weseebothmin{Ps,Fs};Ps+Fs=min{Ps,Fs}wheremin functioniswelldefined. Forif

QuasiSuperMatrices191

    

       

    

       

        = 00.10.200.200.1 0.300.5000.31 00.3000.10.20 000.70.40.10.10.7 0.20.30.30.30.10.70.7 0.40.10.60000.5           Thusonlywhencompatibilityexistswemaybeina

NowweproceedontodefinemaxfunctiononF

Fs=

12 ijij 345 ijijij aa aaa

whereAt=(taij);t=1,2,3,4,5. and Ps=

12 ijij 345 ijijij bb bbb

whereBp=(p ij b);p=1,2,3,4,5. min{Fs+Ps}= 1122 ijijijij 334455 ijijijijijij min{a,b}min{a,b} min{a,b}min{a,b}max{a,b}

positiontodefineminfunction.

sandPs

max{Fs+Ps}= 1122 ijijijij 334455 ijijijijijij

max{a,b}max{a,b} max{a,b}max{a,b}max{a,b}     =

0.30.710.70.610.7 10.20.810.50.51 0.10.910.110.20.3 0.90.30.80.510.11 10.70.510.511 0.50.80.910.30.20.5

         

NowweproceedontodefinemaxminofFs,Ps. i.e.,maxmin{Fs,Ps} = 1122 334455

maxmin{A,B}maxmin{A,B} maxmin{A,B}maxmin{A,B}maxmin{A,B}    maxmin{Fs,Ps}=

                    Attimeswemayhaveonly‘addition’tobecompatible.For instanceconsidertheexample. Example 4.27: LetFsandPsbeanytwoquasifuzzysuper matricesgivenby

0.10.30.30.20.60.70.7 0.20.710.50.50.21 0.30.90.30.30.20.20.3 0.40.30.610.50.41 0.90.30.70.30.50.70.7 0.50.30.60.300.20.5

192NeutrosophicSuperMatricesandQuasiSuperMatrices

Fs=

0.30.810.4100.9 00.40.50.800.80.4 0.8100.60.710.3 0.40.30.80.10.50.60.1 0.50.80.6100.80.1 0.5100.50.610 100.810.400.2 0.40.510.310.40.7

             and Ps=

0.70.30.50.30.50.10 10.80.40.800.40.9 0.70.60.30.10.80.10.3 0.30.80.50.70.90.70.2 0.10.50.10.30.40.50.3 0.30.20.40.70.50.10.8 0.10.80.30.30.70.20.1 0.80.70.10.10.60.30.5

. AaAa AaAa

          and Ps=      12 1ij2ij 34 3ij4ij

          maxmin{Fs,Ps}isnotdefinedasmaxmin{A1,B1}isnot definedminormaxisdefinedby

BbBb BbBb

QuasiSuperMatrices193

             Letusdenote Fs=      12 1ij2ij 34 3ij4ij

max{Fs+Ps}= 1122 ijijijij 3344 ijijijij

max(a,b)max(a,b) max(a,b)max(a,b)         =

0.70.810.410.10.9 10.80.50.800.80.9 0.810.30.60.810.3 0.40.80.80.70.90.70.2 0.50.80.610.40.80.3 0.510.40.70.610.8 10.80.810.70.20.2 0.80.710.310.40.7

            

Thusweseeattimesmaxandminmaybedefinedbutmaxmin maynotbedefined.

Herealsoentriesofallthesequasisupermatricescanbe replacedbytheentriesfromneutrosophicring,dualnumber ring,dualneutrosophicnumberringsandsoon.

Thesenewstructuresfindapplicationsinconstructingthe fuzzyneutrosophicquasisupermodels.

194NeutrosophicSuperMatricesandQuasiSuperMatrices

ABOUT THE AUTHORS

Dr.W.B.Vasantha Kandasamy is an Associate Professor in the Department of Mathematics, Indian Institute of Technology Madras, Chennai. In the past decade she has guided 13 Ph.D. scholars in the different fields of non-associative algebras, algebraic coding theory, transportation theory, fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. She has to her credit 646 research papers. She has guided over 68 M.Sc. and M.Tech. projects. She has worked in collaboration projects with the Indian Space Research Organization and with the Tamil Nadu State AIDS Control Society. She is presently working on a research project funded by the Board of Research in Nuclear Sciences, Government of India. This is her 74th book.

On India's 60th Independence Day, Dr.Vasantha was conferred the Kalpana Chawla Award for Courage and Daring Enterprise by the State Government of Tamil Nadu in recognition of her sustained fight for social justice in the Indian Institute of Technology (IIT) Madras and for her contribution to mathematics. The award, instituted in the memory of Indian-American astronaut Kalpana Chawla who died aboard Space Shuttle Columbia, carried a cash prize of five lakh rupees (the highest prize-money for any Indian award) and a gold medal. She can be contacted at vasanthakandasamy@gmail.com Web Site: http://mat.iitm.ac.in/home/wbv/public_html/ or http://www.vasantha.in

Dr. Florentin Smarandache is a Professor of Mathematics at the University of New Mexico in USA. He published over 75 books and 200 articles and notes in mathematics, physics, philosophy, psychology, rebus, literature. In mathematics his research is in number theory, non-Euclidean geometry, synthetic geometry, algebraic structures, statistics, neutrosophic logic and set (generalizations of fuzzy logic and set respectively), neutrosophic probability (generalization of classical and imprecise probability). Also, small contributions to nuclear and particle physics, information fusion, neutrosophy (a generalization of dialectics), law of sensations and stimuli, etc. He got the 2010 Telesio-Galilei Academy of Science Gold Medal, Adjunct Professor (equivalent to Doctor Honoris Causa) of Beijing Jiaotong University in 2011, and 2011 Romanian Academy Award for Technical Science (the highest in the country). Dr. W. B. Vasantha Kandasamy and Dr. Florentin Smarandache got the 2011 New Mexico Book Award for Algebraic Structures. He can be contacted at smarand@unm.edu