Skew Centrosymmetric Matrices

International Journal of Pure and Applied Mathematical Sciences. ISSN 0972-9828 Volume 10, Number 1 (2017), pp. 99-106 © Research India Publications http://www.ripublication.com

Properties of k - CentroSymmetric and k – Skew CentroSymmetric Matrices

1Dr. N.Elumalai and 2Mrs.B.Arthi

1Associate Professor,2Assistant Professor 1 Department of Mathematics, A.V.C. College (Autonomous), Mannampandal, TamilNadu, India. 2 Department of Mathematics, A.V.C. College (Autonomous), Mannampandal, TamilNadu, India.

Abstract

The basic concepts and theorems of k- Centrosymmetric, k- Skew Centrosymmetric matrices are introduced with examples.

Keywords: Symmetric matrix, Centrosymmetric ,k- Centrosymmetric matrix,Skewsymmetric matrix Skew Centrosymmetric matrix and k-Skew centrosymmetric matrix.

AMS CLASSIFICATIONS: 15B05, 15A09

I. INTRODUCTION The concept of k-symmetric matrices and was introduced in [1], [2] and [3] Some properties of symmetric matrices given in [5],[6] ,[7] .In this paper, our intention is to define k- Centrosymmetric matrix, k-Skew Centrosymmetric matrix and also we discussed some results on Centrosymmetric matrices.

II. PRELIMINARIES AND NOTATIONS C is centrosymmetric matrix,CT is called Transpose of C .Let k be a fixed product of disjoint transposition in Sn and ‘K’be the permutation matrix associated with K.Clearly K satisfies the following properties. K2 = I , KT = K .

100 Dr. N.Elumalai and Mrs. B. Arthi

III. DEFINITIONS AND THEOREMS DEFINITION:1 T A Square matrix A = [aij ] nxn is said to be symmetric if A = A (ie) aij = aji ∀ 푖, 푗

DEFINITION:2 A Square matrix which is symmetric about the centre of its array of elements is called centrosymmetric thus C = [aij ] nxn centrosymmetric if aij = an-i+1,n-j+1.

DEFINITION: 3 A centrosymmetric matrix C Rn xn is called k-centosymmetric matrix if C = K CT K.

THEOREM: 1

Let C  Rn x n is k-centrosymmetric matrix then CT = K C K. Proof: K C K = K CT K where C = CT = CT K K where K CT = C T K = CT K2 = CT

THEOREM:2

If C1 and C2 are k-centrosymmetric matrices then C1C2 is also k-cetrosymmetric matrix Proof : T T Let C1 , and C2 are k-centrosymmetric matrices if C1 = K C1 K and C2 = K C2 K . T T T T Since C1 and C2 are also k-centrosymmetric matrices then C1 = K C1K and C2 =

K C2 K.

To prove C1 C2 is k-centrosymmetric matrix T We will show that C1 C2= K (C1 C2) K T T T Now K (C1 C2) K = K C2 C1 K T T = K [(K C2 K)(K C1 K)]K where C1 = K C1 K and C2 = K C2 K. Properties of k - CentroSymmetric and k – Skew CentroSymmetric Matrices 101

2 2 2 = K C2 K C1 K

=C2 C1

= C1 C2 where C2 C1 = C1 C2 THEOREM :3 If C is k-centro symmetric matrices and K is the permutation matrix, k = (1 2) then KC is also k-centro symmetric matrix. Proof :

A matrix C  Rn x n is said to be k-centrosymmetric matrix if C = K CT K Since CT is also k-centrosymmetric matrices then CT = K C K To prove K C is K- centrosymmetric matrix We will show that KC= K (KC)TK Now K (KC)T K = K ( CT KT ) K where ( KC )T = CT KT = KCT where KT K = I = KC where KCT = KC THEOREM: 4 If C  Rn xn is k-centrosymmetric matrix then C CT is also k-centrosymmetric matrix

Proof : A matrix C  Rn xn is said to be k-centrosymmetric matrix if C = K CT K

Since CT is also k-centrosymmetric matrices then CT = K CK We will show that , C CT = K (C CT ) T K For that, K (C CT ) T K = K [ (CT )T CT ] K where ( KC )T = CT KT = K (C CT) K where (CT )T = C = (C CT) KK where KC = CK = (C CT) K2 where KK=K2 = (C CT ) THEOREM:5 If C  Rn xn is k-centrosymmetric matrix then C ± CT is also k-centrosymmetric matrix 102 Dr. N.Elumalai and Mrs. B. Arthi

Proof : A matrix C Rn xn is said to be k-centrosymmetric matrix if C = K CT K

Since CT is also k-centrosymmetric matrices then CT = K CK We will show that , C +CT = K (C +CT ) T K

T T T T T T T T For that, K (C +C ) K = K [ (C ) +C ] K where ( C1 +C2 ) = ( C1 +C2 ) = K (C +CT) K where (CT )T = C = (C + CT ) KK where KC = CK = (C + CT) K2 = (C +CT )

THEOREM:6

If C1 and C2 are k-centrosymmetric matrices then C 1 ± C2 is also k- cetrosymmetric matrix Proof : T T Let C1 , and C2 are k-centrosymmetric matrices if C1 = K C1 K and C2 = K C2 K . T T T T Since C1 and C2 are also k-centrosymmetric matrices then C1 = K C1K and C2 =

K C2 K.

To prove C1 + C2 is k-centrosymmetric matrix T We will show that C1 + C2= K (C1 + C2) K T T T Now K (C1 + C2) K = K(C1 + C2 ) K T T = K C1 K + K C2 K where C1 = K C1 K and C2 = K C2 K.

= C1 + C2

RESULT:

Let C1 and C2 are k-centrosymmetric matrices for the following conditions are holds

[i] C1 C2 = C2 C1 T T [ii] (C1 C2 C1) and (C2 C1 C2) are also k- centrosymmetric matrices.

[iii] Adj C1 also k-centro symmetric matrix.

[iv] C1(Adj C1) is also k-centrosymmetric matrix. Properties of k - CentroSymmetric and k – Skew CentroSymmetric Matrices 103

EXAMPLE: 1 2 3 1 2 0 1 Let C1 =( ) and C2 =( ) ; K = ( ) 3 2 2 1 1 0 T 0 1 2 3 1 2 0 1 8 7 (i) K (C1 C2 ) K =( ) ( ) ( ) ( ) = ( ) = C1 C2 1 0 3 2 2 1 1 0 7 8 T T 0 1 2 3 1 2 2 3 0 1 (ii) K (C1 C2 C1) K ==( ) ( ) ( ) ( ) ( ) = 1 0 3 2 2 1 3 2 1 0 37 38 T ( ) = C1 C2 C1 38 37

DEFINITION:4 T A Square matrix A = [aij ] nxn is said to be skewsymmetric matrix if A = - A (ie) aij = - aji ∀ 푖, 푗

DEFINITION:5 T A Square matrix C = [aij ] nxn is called skew centrosymmetric matrix if C = - C

DEFINITION: 6 A skew centrosymmetric matrix C Rn xn is called k-skew centosymmetric matrix if K C K = -CT.

THEOREM: 7

Let C Rn x n is k- skew centrosymmetric matrix then K CT K. = - C Proof: K CT K = K (-C ) K = where CT = -C = - C K K = -C THEOREM:8

If C1 and C2 are k- skew centrosymmetric matrices then C1C2 is also k- skew cetrosymmetric matrix Proof : 104 Dr. N.Elumalai and Mrs. B. Arthi

T T Let C1 , and C2 are k-skew centrosymmetric matrices if K C1 K = - C1 and K C2 K = -C2 . T T T Since C1 and C2 are also k- skew centrosymmetric matrices then K C1K = - C1 T and K C2 K = - C2 .

To prove C1 C2 is k- skew centrosymmetric matrix T We will show that C1 C2= K (C1 C2) K T T T Now K (C1 C2) K = K C2 C1 K T T = K [( -K C2 K)( -K C1 K)]K where K C1 K = - C1 and K C2 K == - C2 . 2 2 2 = K C2 K C1 K 2 =C2 C1 where K = I

= C1 C2 where C2 C1 = C1 C2

THEOREM :9 If C is k-skew centrosymmetric matrix and K is the permutation matrix, k = (1 2) then - KC is also k- skew centro symmetric matrix. Proof :

A matrix CRn x n is said to be k- skew centrosymmetric matrix if K CT K = -C Since CT is also k-skew centrosymmetric matrices then K C K = - CT To prove - K C is K- skew centrosymmetric matrix We will show that -KC= K (KC)TK Now K (KC)T K = K ( CT KT ) K where ( KC )T = CT KT = KCT where KT K = I = - KC where KCT = - KC

THEOREM: 10 If C Rn xn is k-skew centrosymmetric matrix then C CT is also k- skew centrosymmetric matrix Proof : A matrix C  Rn xn is said to be k-skew centrosymmetric matrix if K CT K = -C Properties of k - CentroSymmetric and k – Skew CentroSymmetric Matrices 105

Since CT is also k- skew centrosymmetric matrices then K CK = - CT We will show that , C CT = K (C CT ) T K For that, K (C CT ) T K = K [ (CT )T CT ] K where ( KC )T = CT KT = K (C CT) K where (CT )T = C = (C CT) KK where KC = CK = (C CT) K2 where KK=K2 = (C CT ) RESULT: 1. If C  Rn xn is k-skew centrosymmetric matrix then C – C T is also k- skew centrosymmetric matrix.

2. Let C1 and C2 are k- skewcentrosymmetric matrices for the following conditions are holds

[i] C1 C2 = C2 C1 T T [ii] (C1 C2 C1) and (C2 C1 C2) are also k- skew centrosymmetric matrices

[iii] Adj C1 also k- skew centro symmetric matrix

[iv] C1(Adj C1) is also k- skew centrosymmetric matrix .

REFERENCES [1] Ann Lec. Secondary symmetric and skew symmetric secondary orthogonal matrices; (i) Period, Math Hungary, 7, 63-70(1976). [2] A. Cantoni and P. Butler, Eigenvalues and eigenvectors of symmetric centrosymmetric matrices, Linear Algebra Appl. 13 (1976), 275–288. [3] James R. Weaver, Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors, Amer. Math. Monthly 92 (1985), 711–717. [4] Gunasekaran.K, Mohana.N,” k-symmetric Double stochastic , s-symmetric Double stochastic , s-k-symmetric Double stochastic Matrices” International Journal of Engineering Science Invention, Vol 3,Issue 8,(2014) . [5] Hazewinkel, Michiel, ed. (2001), "Symmetric matrix", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 . [6] Krishnamoorthy.S, Gunasekaran.K, Mohana.N, “ Characterization and theorems on doubly stochastic matrices” Antartica Journal of Mathematics, 11(5)(2014). 106 Dr. N.Elumalai and Mrs. B. Arthi

[7] Elumalai .N,Rajesh kannan .K “ k - Symmetric Circulant, s - Symmetric Circulant and s – k Symmetric Circulant Matrices “ Journal of ultra scientist of physical sciences, Vol.28 (6),322-327 (2016 ).