Properties of k - CentroSymmetric and k Skew CentroSymmetric Matrices

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1 International Journal of Pure and Applied Mathematical Sciences. ISSN Volume 10, Number 1 (2017), pp Research India Publications Properties of k - CentroSymmetric and k Skew CentroSymmetric Matrices 1 Dr. N.Elumalai and 2 Mrs.B.Arthi 1 Associate Professor, 2 Assistant 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,c T 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. K 2 = I, K T = K.

2 100 Dr. N.Elumalai and Mrs. B. Arthi III. DEFINITIONS AND THEOREMS DEFINITION:1 A Square matrix A = [aij ] nxn is said to be symmetric if A = A T (ie) aij = aji i, j 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 R n xn is called k-centosymmetric matrix if C = K C T K. THEOREM: 1 Let C R n x n is k-centrosymmetric matrix then C T = K C K. Proof: K C K = K C T K where C = C T = C T K K where K C T = C T K = C T K 2 = C T THEOREM:2 If C1 and C2 are k-centrosymmetric matrices then C1C2 is also k-cetrosymmetric matrix Let C1, and C2 are k-centrosymmetric matrices if C1 = K C1 T K and C2 = K C2 T K. Since C1 T and C2 T are also k-centrosymmetric matrices then C1 T = K C1K and C2 T = K C2 K. To prove C1 C2 is k-centrosymmetric matrix We will show that C1 C2= K (C1 C2) T K Now K (C1 C2) T K = K C2 T C1 T K = K [(K C2 K)(K C1 K)]K where C1 T = K C1 K and C2 T = K C2 K.

3 Properties of k - CentroSymmetric and k Skew CentroSymmetric Matrices 101 THEOREM :3 = K 2 C2 K 2 C1 K 2 =C2 C1 = C1 C2 where C2 C1 = C1 C2 If C is k-centro symmetric matrices and K is the permutation matrix, k = (1 2) then KC is also k-centro symmetric matrix. A matrix C R n x n is said to be k-centrosymmetric matrix if C = K C T K Since C T is also k-centrosymmetric matrices then C T = K C K To prove K C is K- centrosymmetric matrix We will show that KC= K (KC) T K Now K (KC) T K = K ( C T K T ) K where ( KC ) T = C T K T THEOREM: 4 = KC T where K T K = I = KC where KC T = KC If C R n xn is k-centrosymmetric matrix then C C T is also k-centrosymmetric matrix A matrix C R n xn is said to be k-centrosymmetric matrix if C = K C T K Since C T is also k-centrosymmetric matrices then C T = K CK We will show that, C C T = K (C C T ) T K For that, THEOREM:5 K (C C T ) T K = K [ (C T ) T C T ] K where ( KC ) T = C T K T = K (C C T ) K where (C T ) T = C = (C C T ) KK where KC = CK = (C C T ) K 2 where KK=K 2 = (C C T ) If C R n xn is k-centrosymmetric matrix then C ± C T is also k-centrosymmetric matrix

4 102 Dr. N.Elumalai and Mrs. B. Arthi A matrix C R n xn is said to be k-centrosymmetric matrix if C = K C T K Since C T is also k-centrosymmetric matrices then C T = K CK We will show that, C +C T = K (C +C T ) T K For that, K (C +C T ) T K = K [ (C T ) T +C T ] K where ( C 1 +C 2 ) T T = ( C 1 +C T 2 ) = K (C +C T ) K where (C T ) T = C = (C + C T ) KK where KC = CK = (C + C T ) K 2 = (C +C T ) THEOREM:6 If C1 and C2 are k-centrosymmetric matrices then C 1 ± C2 is also k- cetrosymmetric matrix Let C1, and C2 are k-centrosymmetric matrices if C1 = K C1 T K and C2 = K C2 T K. Since C1 T and C2 T are also k-centrosymmetric matrices then C1 T = K C1K and C2 T = K C2 K. To prove C1 + C2 is k-centrosymmetric matrix We will show that C1 + C2= K (C1 + C2) T K Now K (C1 + C2) T K = K(C1 T + C2 T ) K = K C1 K + K C2 K where C1 T = K C1 K and C2 T = K C2 K. = C1 + C2 RESULT: Let C1 and C2 holds are k-centrosymmetric matrices for the following conditions are [i] C 1 C 2 = C 2 C 1 [ii] (C T 1 C 2 C 1 ) and (C T 2 C 1 C 2 ) are also k- centrosymmetric matrices. [iii] Adj C 1 also k-centro symmetric matrix. [iv] C 1 (Adj C 1 ) is also k-centrosymmetric matrix.

5 Properties of k - CentroSymmetric and k Skew CentroSymmetric Matrices 103 EXAMPLE: 1 Let C 1 =( ) and C 2 =( ) ; K = ( ) (i) K (C 1 C 2 ) T K =( ) ( ) ( ) ( ) = ( ) = C 1 C 2 (ii) K (C T 1 C 2 C 1 ) T K ==( ) ( ) ( ) ( ) ( ) = ( ) = C 1 T C 2 C 1 DEFINITION:4 A Square matrix A = [aij ] nxn is said to be skewsymmetric matrix if A = - A T (ie) aij = - aji i, j DEFINITION:5 A Square matrix C = [aij ] nxn is called skew centrosymmetric matrix if C = - C T DEFINITION: 6 A skew centrosymmetric matrix C R n xn is called k-skew centosymmetric matrix if K C K = -C T. THEOREM: 7 Let C R n x n is k- skew centrosymmetric matrix then K C T K. = - C Proof: THEOREM:8 K C T K = K (-C ) K = where C T = -C = - C K K = -C If C1 and C2 are k- skew centrosymmetric matrices then C1C2 is also k- skew cetrosymmetric matrix

6 104 Dr. N.Elumalai and Mrs. B. Arthi Let C1, and C2 are k-skew centrosymmetric matrices if K C1 T K = - C1 and K C2 T K = -C2. Since C1 T and C2 T are also k- skew centrosymmetric matrices then K C1K = - C1 T and K C2 K = - C2 T. To prove C1 C2 is k- skew centrosymmetric matrix We will show that C1 C2= K (C1 C2) T K Now K (C1 C2) T K = K C2 T C1 T K = K [( -K C2 K)( -K C1 K)]K where K C1 K = - C1 T and K C2 K == - C2 T. = K 2 C2 K 2 C1 K 2 =C2 C1 where K 2 = 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. A matrix C R n x n is said to be k- skew centrosymmetric matrix if K C T K = -C Since C T is also k-skew centrosymmetric matrices then K C K = - C T To prove - K C is K- skew centrosymmetric matrix We will show that -KC= K (KC) T K Now K (KC) T K = K ( C T K T ) K where ( KC ) T = C T K T = KC T where K T K = I = - KC where KC T = - KC THEOREM: 10 If C R n xn is k-skew centrosymmetric matrix then C C T is also k- skew centrosymmetric matrix A matrix C R n xn is said to be k-skew centrosymmetric matrix if K C T K = -C

7 Properties of k - CentroSymmetric and k Skew CentroSymmetric Matrices 105 Since C T is also k- skew centrosymmetric matrices then K CK = - C T We will show that, C C T = K (C C T ) T K For that, RESULT: K (C C T ) T K = K [ (C T ) T C T ] K where ( KC ) T = C T K T = K (C C T ) K where (C T ) T = C = (C C T ) KK where KC = CK = (C C T ) K 2 where KK=K 2 = (C C T ) 1. If C R n 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] C 1 C 2 = C 2 C 1 [ii] (C 1 T C 2 C 1 ) and (C 2 T C 1 C 2 ) are also k- skew centrosymmetric matrices [iii] Adj C 1 also k- skew centro symmetric matrix [iv] C 1 (Adj C 1 ) 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), [3] James R. Weaver, Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors, Amer. Math. Monthly 92 (1985), [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 [6] Krishnamoorthy.S, Gunasekaran.K, Mohana.N, Characterization and theorems on doubly stochastic matrices Antartica Journal of Mathematics, 11(5)(2014).

8 106 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), (2016 ).