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Characterizations of Con-K-Normal Centrosymmetric Matrix

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Characterizations of Con-K-Normal Centrosymmetric Matrix International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-06, Sep 2018 Characterizations Of Con-K-Normal Centrosymmetric Matrix 1Dr.N.Elumalai, 2Mrs.B.Arthi 1Associate professor, 2Assistant professor, A.V.C. College (Autonomous), Mannampandal, TamilNadu, India. Abstract: The basic concepts and theorems of CON-K-Normal Centrosymmetric matrices are introduced with examples. Keywords: Centrosymmetric,Normalcentrosymmetric,con-K-Normal centrosymmetric matrix. AMS Subject Classification: 15A09,15B05,15B99. I. INTRODUCTION Definition:2 A centrosymmetric matrix A is said to be normal For a number of decades symmetric matrices over the real centrosymmetric If A = A. field have been studied intently by every beginning linear algebra student as aclass of matrices which are defined by Definition:3 the property of being symmetric about their main diagonal A centrosymmetric matrix A is said to be K-normal or by the fact that you can interchange the rows and centrosymmetric matrix If A K = K A. columns of a given symmetric matrix and it remains Definition:4 unchanged.In [1] A.C.Aitken defines a centrosymmetric matrix P and this definition coincides with the definition A centrosymmetric matrix A is said to be con K- given by Graybill of a centrosymmetric matrix[6] normal centrosymmetric matrix. In this paper we will discuss about the basic properties and If A =K charachracterisation of con-K-Normal Centrosymmetric Theorem:1 matrices, also we will discuss some results on For A the following conditions are equivale centrosymmetric matrices. a)A is con k-normal centrosymmetric Let Cnxn denote the set of all nxn complex matrix. A is b) is con k-normal centrosymmetric con-K-Normal centrosymmetric matrix. A* is calles c) is con k-normal centrosymmetric. conjugate transpose of A . Let K be a fixed product of d) is con k-normal centrosymmetric. disjoint transposition in Sn and K be the permutation e) is con k-normal centrosymmetric,if is exist. matrix associated with K. Clearly K satisfies the following f) is con k-normal centrosymmetric. properties. g)Λais con k-normal centrosymmetric, where λ is real 2 T K = I, K =K. number. II. DEFINITIONS AND THEOREMS Proof: (a)(b) Definition:1 A is con k-normal centrosymmetric. A square matrix which is symmetric about the centre of its A K=K array of elements is called centrosymmetric thus = C= centrosymmetric if, is con K-normal centrosymmetric. = (a)(c) 316 | IJREAMV04I0642078 DOI : 10.18231/2454-9150.2018.0736 © 2018, IJREAM All Rights Reserved. International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-06, Sep 2018 A is con K-normal centrosymmetric. Similatrly, we can prove (-iA) is con K-normal A K=K centrosymmetric. = Theorem:3 Pre and post multiply by K on both sides, Let A be con K-normal centrosymmetric.Then A K=K A and A are K-normal centrosymmetric is con K-normal centrosymmetric. (a)(d) Proof Let A be con K-normal centrosymmetric. A is con K-normal centrosymmetric. If A K=K . A K=K We have, = (A ) K=(A )( )K Pre and post multiply by K on both sides, =A( ) K A K=K = (3) is con K-normal centrosymmetric. And (a)(e) K (A )=K( )( A) A is con K-normal centrosymmetric. =[K( )]( A) A K=K = (4) = From (3) and (4), we have is con K-normal centrosymmetric. (A ) K=K (A ) (a)(f) A is k-normal centrosymmetric. A is con K-normal centrosymmetric. Similarly, A K=K We can prove, is K-normal centrosymmetric. = Remark: is con K-normal centrosymmetric. The reverse implication A and A are K-normal (a)(g) centrosymmetric,then A is con K-normal centrosymmetric A is con K-normal centrosymmetric. is false. A K=K Theorem:4 (A K)= (K λA is con k-normal centrosymmetric. Let A,B be con K-normal centrosymmetric matrices. Then,A =B B Theorem:2 In otherwords, If A con-commutes with some matrix B. Let A Then con commutes with B as well a)If A is con K-normal centrosymmetric.Then (iA) is con Proof: K-centrosymetric. b)If A is con K-normal centrosymmetric,Then (-iA) is con Given A and B are con K-normal centrosymmetric. K-normal centrosymmetric. Then, A K=K B K=K Proof: Since, A =B is trivially true for B=A. Let A be con K-normal centrosymmetric and let B be con (a)Given A is con K- normal centrosymmetric. commute with A. i.e.A K=K A =B (A K)= (K ) For ̂=[ ] and ̂=[ ] iA is con k-normal centrosymmetric. 317 | IJREAMV04I0642078 DOI : 10.18231/2454-9150.2018.0736 © 2018, IJREAM All Rights Reserved. International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-06, Sep 2018 We have, ̂ ̂=A B matrices; International journal of Mathematics Trends ̂ ̂=B A and Technology, Vol.4,Issue.11, PP.369-385(2013). [7] F.Graybill Introduction to matrices with application in ̂ and ̂commutes static,Wadsworth,Belmort CA.1969. Since A is con K-normal centrosymmetric, then is con [8] Elumala.N., Arthi.B.,:Properties of k-centrosymmetric K-normal centrosymmetric. and k-skew centrosymmetric matrices. International ̂ is con K-normal centrosymmetric. journal of pure and Applied Mathematical Science ̂ is con K- normal centrosymmetric ̂ ̂ ̂ ̂ Volume-10, 99-106; July,(2017). Which is equivalent to =B Example: 3 2 i 3 2 i * 1.Let A= 4 i 4 A = 4 i 4 i 2 3 i 2 3 0 0 1 k = 0 1 0 then 1 0 0 8 6 4i 18 * AA K= 12 8i 17 12 8i 18 6 4i 8 =KAT REFERENCE [1] A.C. Boud,Edin Aitken,Determinat and matrices,Oliver and Boyd,Edinburgh 1939 [2] Ann Lec; Secondary symmetric and skew symmetric secondary orthogonal matrices; (i) period,Math Hungary,7, 63-70(1976). [3] Cantoni and P.Butler; Eigenvalues and Eigen vectors of symmetric centrosymmetric matrices, Linear Algebra Appl. 13(1976), 275 – 288. [4] James R. Weaver; Centro symmetric (cross- symmetric) matrices, their basic properties, Eigen values and Eigen vectors, Amer. Math, Monthly 92(1985), 711 – 717. [5] Krishnamoorthy.S.,Gunasekaran.K., and Arumugam.K. : Products of Conjugate k-normal matrices; International journal of Mathematics Trends and Technology, Vol.4,Issue.11, PP.369-385(2013). [6] Krishnamoorthy.S.,Gunasekaran.K., and Arumugam.K. : Products of Conjugate k-normal 318 | IJREAMV04I0642078 DOI : 10.18231/2454-9150.2018.0736 © 2018, IJREAM All Rights Reserved. .
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