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.