Equivalent Conditions on Conjugate Unitary Matrices
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1 International Journal of Mathematics Trends and Technology (IJMTT Volume 42 Number 1- February 2017 Equivalent Conditions on Conjugate Unitary Matrices A. Govindarasu #, S.Sassicala #1 # Department of Mathematics, A.V.C. College (Autonomous Mannampandal, Mayiladuthurai Abstract Concept of conjugate unitary matrices are given. Some equivalent conditions on conjugate unitary matrices are obtained. Conditions related to secondary unitary matrices, hermitian, s-hermitian and involutary are also derived. AMS Classification 15A09, 15A57. Keywords Unitary matrix, secondary transpose of a matrix, conjugate secondary transpose of a matrix, conjugate unitary matrix. I. INTRODUCTION Anna Lee [1] has initiated the study of secondary symmetric matrices. Also she has shown that for a Complex matrix A, the usual transpose and secondary transpose are related as, where is the permutation matrix with units in its secondary diagonal. Also denotes the conjugate secondary transpose of. where [2]. In this paper our intension is to prove some equivalent conditions on conjugate unitary matrices. II. PRELIMINARIES AND NOTATIONS Let be the space of complex matrices of order n. For.,,,, denote transpose, conjugate, conjugate transpose, secondary transpose, conjugate secondary transpose of a matrix respectively. Also satisfies and is called unitary if [7]. is called conjugate normal if [3]. is called s-normal iff [6] is called secondary unitary (s-unitary if [5]. is called s-hermitian if A A A S. is called is called conjugate unitary if [4]. III. EQUIVALENT CONDITIONS ON CONJUGATE UNITARY MATRICES Theorem 3.1 Any two of the following imply the (i is conjugate unitary matrix (ii is hermitian (iii Proof (i and (ii (iii A Case (i ( is hermitian Case (ii ( is hermitian (ii and (iii (i (. ( is hermitian (3.1.1 Again take ( ( is hermitian (3.1.2 Therefore, from (3.1.1 and (3.1.2 we A is conjugate unitary matrix. (iii and (i (ii Case (i Pre multiplying the above equation by ( A is hermitian Case(ii ISSN: Page 64
2 International Journal of Mathematics Trends and Technology (IJMTT Volume 42 Number 1- February 2017 Post multiplying the above equation by ( A is hermitian Therefore,in both cases implies A is hermitian. Theorem 3.2: Any two of the following imply the (i is conjugate unitary matrix (ii is s-unitary (iii commutes with V ( Proof: (i and (ii (iii Case (i ( is s - unitary Taking conjugate transpose Post multiplying the above equation by V Case (ii Taking conjugate ( is s - unitary Post multiply the above equation by sides, we commutes with (ii and (iii (i is s unitary Taking conjugate transpose (3.2.1 = Taking conjugate transpose (3.2.2 Therefore from equations (3.2.1 and (3.2.2 we, (iii and (i (ii Case (i (3.2.3 Pre multiplying the above equation by Post multiplying equation (3.2.3 by (3.2.4 (3.2.5 Therefore, from (3.2.4 and (3.2.5, we Case (ii (3.2.6 Pre multiplying equation (3.2.6 by sides, we Post multiplying equation (3.2.6 by we (3.2.7 sides, (3.2.8 Therefore from equations (3.2.7 and (3.2.8, we Therefore, in both cases, implies is s-unitary matrix. Theorem 3.3: Any two of the following imply the (i is conjugate unitary matrix (ii is s hermitian (iii ISSN: Page 65
3 International Journal of Mathematics Trends and Technology (IJMTT Volume 42 Number 1- February 2017 Proof: (i and (ii (iii Case (i Case (ii Case (ii A ( ( is s- hermitian (ii and (iii (i ( V A is s- hermitian. Theorem3.4: Any two of the following imply the (i is conjugate unitary matrix (ii is skew hermitian (iii is Involutary Proof : (i and (ii (iii Case (i A (3.3.1 A V ( V Case (ii ( (ii and (iii (i (3.3.2 Therefore from equations (3.3.1 and (3.3.2 we is conjugate unitary matrix. (iii and (i (ii Case (i ( Again take (3.4.1 ISSN: Page 66
4 International Journal of Mathematics Trends and Technology (IJMTT Volume 42 Number 1- February 2017 Case (ii, (3.4.2 Therefore, from (3.4.1 and (3.4.2 we (iii and (ii (i Take (3.5.4, (3.5.5 Post multiplying equation (3.5.4 by sides, we, (3.5.6 Therefore, from equations (3.5.5 and (3.5.6, we, Therefore, in both cases, implies is s-unitary. (iii and (i (ii ( is s unitary Now we take (3.4.3 ( is hermitian (3.5.7 Therefore, from (3.4.3 and (3.4.4 we (3.4.4 ( is hermitian is skew-hermitian Theorem3.5: If then any two of the following imply the (i is s-unitary (ii is conjugate unitary (iii is hermitian Proof: Given that (i and (ii (iii is s- unitary is conjugate unitary is hermitian (ii and (iii (i Case (i (3.5.1 Post multiplying equation (3.5.1 by (3.5.2 ( is hermitian ( is hermitian, (3.5.8 Therefore, from equations (3.5.7 and (3.5.8, we is conjugate unitary matrix. Theorem 3.6: Any two of the following imply the (i is conjugate unitary matrix (ii is hermitian (iii Proof: (i and (ii (iii Case (i Take transpose (3.5.3 Therefore, from equations (3.5.2 and (3.5.3 we ISSN: Page 67
5 International Journal of Mathematics Trends and Technology (IJMTT Volume 42 Number 1- February 2017 Case (ii Again taking transpose Again take (3.6.1 is hermitian Taking transpose Taking transpose Therefore, in both cases we (ii and (iii (i. Again taking transpose Taking transpose (3.6.2 Therefore, from equations (3.6.1 and (3.6.2, we (iii and (ii (i is conjugate unitary matrix. Taking transpose is hermitian IV. CONCLUSION Some equivalent condition on conjugate unitary matrices are obtained. Conditions related to secondary unitary matrices, unitary matrices, hermitian, s-hermitian and involutary matrix are also derived. [6]. S. Krishnamoorthy and R. Vijayakumar, Some equivalent conditions on s-normal matrices, International Journal of Comtemporary Mathematical Sciences, Vol.4, No.29, pp , [7]. A.K. Sharma, Text book of matrix theory, DPH Mathematics series, 2004 REFERENCES [1]. Anna Lee, Secondary symmetric, secondary skew symmetric, secondary orthogonal matrices, Period. Math. Hungary, Vol. 7, pp , [2]. Anna Lee, On s-symmetric, s-skew symmetric and s- orthogonal matrices, Period. Math. Hungary, Vol. 7, pp , [3]. A. Faßbender and Kh.D.Ikramov, Conjugate-normal matrices: a survey, Lin. Alg. Appl., Vol. 429, pp , [4]. A. Govindarasu and S. Sassicala, Characterizations of conjugate unitary matrices, Journal of ultra scientist of physical science, Vol. 29, No. 1, pp , [5]. S. Krishnamoorthy and A. Govindarasu, On Secondary Unitary Matrices, International Journal of computational science and Mathematics, Vol.2, No.3, pp , ISSN: Page 68
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