Smarandache cyclic geometric determinant sequences

Scientia Magna Vol. 8 (2012), No. 4, 88-91

Smarandache cyclic geometric determinant sequences A. C. F. Bueno Department of Mathematics and Physics, Central Luzon State University, Science City of Mu˜ noz 3120, Nueva Ecija, Philippines Abstract In this paper, the concept of Smarandache cyclic geometric determinant sequence was introduced and a formula for its nth term was obtained using the concept of right and left circulant matrices.

Keywords Smarandache cyclic geometric determinant sequence, determinant, right circulan -t matrix, left circulant matrix.

§1. Introduction and preliminaries Majumdar [1] gave the formula for nth term of the following sequences: Smarandache cyclic natural determinant sequence, Smarandache cyclic arithmetic determinant sequence, Smarandache bisymmetric natural determinant sequence and Smarandache bisymmetric arithmetic determinant sequence.

Definition 1.1. A Smarandache cyclic geometric determinant sequence {SCGDS(n)} is a sequence of the form  ¯  ¯   ¯ a {SCGDS(n)} = |a|, ¯¯   ¯ ar 

¯ ¯ ¯ a ¯ ¯ ar ¯ ¯ ¯ , ¯ ar ¯ ¯ a ¯ ¯ ¯ ar2

Definition 1.2. A matrix RCIRCn (~c) ∈ Mnxn (R) is if it is of the form  c0 c1 c2 ...   cn−1 c0 c1 ...    cn−2 cn−1 c0 ...  RCIRCn (~c) =  . .. .. . .  .. . . .    c c3 c4 ...  2 c1 c2 c3 ... where ~c = (c0 , c1 , c2 , ..., cn−2 , cn−1 ) is the circulant vector.

ar ar2 a

¯   ar2 ¯¯   ¯ a ¯¯ , . . . .   ¯  ar ¯

said to be a right circulant matrix

cn−2 cn−3 cn−4 .. . c0 cn−1

cn−1



 cn−2    cn−3   , ..  .    c1   c0

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Smarandache cyclic geometric determinant sequences

Definition 1.3. A matrix LCIRCn (~c) ∈ Mnxn (R) is it is of the form  c0 c1 c2 ...   c1 c2 c3 ...    c2 c3 c4 ...  LCIRCn (~c) =  . . .. . .  .. .. . .    c  n−2 cn−1 c0 ... cn−1

c0

c1

...

said to be a leftt circulant matrix if

cn−2

cn−1

cn−1

c0

c0 .. .

c1 .. .

cn−4

cn−3

cn−4

cn−2

       ,     

where ~c = (c0 , c1 , c2 , ..., cn−2 , cn−1 ) is the circulant vector.

Definition 1.4. A right circulant matrix RCIRCn (~g ) with geometric sequence is a matrix of the form 

a

  arn−1    arn−2  RCIRCn (~g ) =  ..  .    ar2  ar

ar

ar2

...

arn−2

a

ar

...

arn−3

arn−1 .. .

a .. .

... .. .

arn−4 .. .

ar3

ar4

2

3

ar

ar

...

a

...

n−1

ar

arn−1



 arn−2    n−3  ar  . ..  .   ar   a.

Definition 1.5. A left circulant matrix LCIRCn (~g ) with geometric sequence is a matrix of the form   a ar ar2 ... arn−2 arn−1     ar ar2 ar3 ... arn−1 a     3 4  ar2 ar ar ... a ar    LCIRCn (~g ) =  . .. .. .. .. .. ..   . . . . . .      arn−2 arn−1 n−4 n−3  a ... ar ar   n−1 n−4 n−2 ar a ar ... ar ar .

The right and left circulant matrices has the following relationship:

 where Π = 

O2 = O1T .

1 O2

O1 ˜ In−1

LCIRCn (~c) = ΠRCIRCn (~c).  0 0 ... 0     0 0 ... 1  . . .. .  with I˜n−1 =   .. .. . ..    0 1 ... 0  1 0 ... 0

 1

     , O1 = (0 0 0 ... 0) and   0   0 0 .. .

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A. C. F. Bueno

No. 4

Clearly, the terms of {SCGDS(n)} are just the determinants of LCIRCn (~g ). Now, for the rest of this paper, let |A| be the notation for the determinant of a matrix A. Hence {SCGDS(n)} = {|LCIRC1 (~g )| , |LCIRC2 (~g )| , |LCIRC3 (~g )| , . . .} .

§2. Preliminary results Lemma 2.1. |RCIRCn (~g )| = an (1 − rn )n−1 .

Proof. 

a

  arn−1    arn−2  RCIRCn (~g ) =  ..  .    ar2  ar  1   rn−1   n−2  r  = a .  ..    r2  r

ar

ar2

...

arn−2

a

ar

...

arn−3

arn−1 .. .

a .. .

... .. .

arn−4 .. .

ar3

ar4

2

3

ar

ar

...

a

...

n−1

ar

r

r2

...

rn−2

1

r

...

rn−3

rn−1 .. .

1 .. .

... .. .

rn−4 .. .

r3

r4

...

2

3

r

By applying the row operations −rn−k R1 + Rk+1  1 r r2   0 −(rn − 1) −r(rn − 1)    0 0 −(rn − 1)  RCIRCn (~g ) ∼ a . .. ..  .. . .    0 0 0  0 0 0

r

...

1 r

n−1

arn−1



 arn−2    arn−3    ..  .   ar   a 

rn−1

 rn−2    n−3  r  . ..  .    r   1.

→ Rk+1 where k = 1, 2, 3, . . . , n − 1, ... ... ... .. . ... ...

rn−2

rn−1



 −rn−3 (rn − 1) −rn−2 (rn − 1)    n−4 n n−3 n −r (r − 1) −r (r − 1)   . .. ..  . .   n n −(r − 1) −r(r − 1)   0 −(rn − 1)

Since |cA| = cn |A| and its row equivalent matrix is a lower traingular matrix it follows that |RCIRCn (~g )| = an (1 − rn )n−1 .

Lemma 2.2. n−1 |Π| = (−1)b 2 c , where bxc is the floor function.

Proof. Case 1: n = 1, 2, |Π| = |In | = 1.

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Case 2: n is even and n > 2 If n is even then there will be n − 2 rows to be inverted because there are two 1’s in the main diagonal. Hence there will be n−2 2 inversions to bring back Π to In so it follows that n−2 |Π| = (−1) 2 .

Case 3: n is odd and and n > 2 If n is odd then there will be n − 1 rows to be inverted because of the 1 in the main diagonal of the frist row. Hence there will be n−1 inversions to 2 bring back Π to In so it follows that |Π| = (−1) But

¥ n−1 ¦ 2

=

¥ n−2 ¦ 2

n−1 2

.

, so the lemma follows.

§3. Main results Theorem 3.1. The nth term of {SCGDS(n)} is given by n−1 SCGDS(n) = (−1)b 2 c an (1 − rn )n−1

via the previous lemmas.

References [1] M. Bahsi and S. Solak, On the Circulant Matrices with Arithmetic Sequence, Int. J. Contemp. Math. Sciences, 5, 2010, No. 25, 1213 - 1222.

[2] H. Karner, J. Schneid, C. Ueberhuber, Spectral decomposition of real circulant matrices, Institute for Applied Mathematics and Numerical Analysis, Vienna University of Technology, Austria.

[3] A. A. K. Majumdar, On some Smarandache determinant sequence, Scientia Magna, 4 (2008), No. 2, 89-95.