Some Properties of Conjugate Unitary Matrices

Transcription

1 Volume 119 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu Some Properties of Conjugate Unitary Matrices A.Govindarasu and S.Sassicala PG and Research Department of Mathematics, A.V.C.College(Autonomous) Mannampandal, Mayiladuthurai, Tamil Nadu, India. sasram10@gmail.com 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. The concept of unitary similarity between two complex n n matrices is extended to unitary similarity and s-unitary similarity between two conjugate unitary matrices, are also introduced and some results are obtained. AMS Subject Classification: 15A09, 15A57, 15A24. Key Words and Phrases: unitary matrix, s-unitary matrix,secondary transpose of a matrix, conjugate secondary transpose of a matrix, conjugate unitary matrix. 1 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 A T and secondary transpose A s are related as A s = V A T V, where V is the permutation matrix with units in its secondary diagonal. Also A s denotes the conjugate secondary transpose of A. i.e., A s = (c ij ) where c ij = a n j+1,n i+1 [2]. Any square matrix over a field is similar to its transpose and any 75

2 square complex matrix is similar to a symmetric complex matrix. Orthogonal similarity of a real matrix and its transpose was obtained by J.Vermeer [8]. In this paper our intension is to prove some equivalent conditions, unitary similarity and s-unitary similarity on conjugate unitary matrices. 1.1 Notations Let C n n be the space of n n complex matrices of order n. For A C n n. Let A T, A, A, A s, A s (= A θ ) denote transpose, conjugate, conjugate transpose, secondary transpose, conjugate secondary transpose of a matrix A respectively. Also the permutation matrix V satisfies V T = V = V = V and V 2 = I. A matrix A C n n is called unitary if AA = [7] A matrix A C n n is called conjugate normal if AA = A A [3] A matrix A C n n is called s-normal if and only if AA θ = A θ A [6] A matrix A C n n is called secondary unitary (s-unitary) if AA θ = A θ A = I [5] A matrix A C n n is called s-hermitian if A = A θ = A s A matrix A C n n is called conjugate unitary if AA = [4] 2 Some Properties of Conjugate Unitary Matrices Theorem 1. If A is conjugate unitary matrix then secondary transpose of A is conjugate unitary matrix. Proof. Assume that A is conjugate unitary matrix. i.e., AA = To show A s (A s ) = (A s ) A s = I AA = I 76

3 Taking secondary transpose on both sides, we have (AA ) s = I s (A ) s A s = I ( I s = I) (A s ) A s = I (A s ) A s = I (A s ) A s = I ( I = I) (1) Case (ii): ( I = I) Taking secondary transpose on both sides we have (A A) s = I s A s (A ) s = I ( I s = I) A s (A s ) = I (2) Therefore, from equations (1) and (2), we have A s (A s ) = (A s ) A s = I Secondary transpose of A is conjugate unitary matrix. Theorem 2. If A is conjugate unitary matrix then V A is conjugate unitary matrix. Proof. A is conjugate unitary matrix AA = (V A) (V A) = V AA V = V AA V ( V = V ) = V V ( A is conjugate unitary) = V 2 = I ( V 2 = I) 77

4 Case (ii): (V A) (V A) = A V V A = A V V A ( V = V ) = A V 2 A = A A ( V 2 = I) = I ( A is conjugate unitary) (V A) (V A) = I (V A) (V A) = I ( I = I) Therefore, in both cases, we have (V A) (V A) = (V A) (V A) = I V A is conjugate unitary matrix. Theorem 3. Any two of the following imply the other. (i). A is conjugate unitary (ii). A is hermitian (iii). A s is involutary Proof. (i) and (ii) (iii) A is conjugate unitary AA = AA = I Taking secondary transpose on both sides, we have (AA ) s = I s (A ) s A s = I ( I s = I) A s A s = I (A s ) 2 = I ( A is hermitian) Case (ii): Taking conjugate transpose on both sides, we have ( I = I) 78

5 Taking secondary transpose on both sides, we have (A A) s = I s A s (A ) s = I ( I s = I) A s A s = I (A s ) 2 = I ( A is hermitian) Therefore, in both cases, we have (A s ) 2 = I A s is involutary. (ii) and (iii) (i) (A s ) 2 = I A s A s = I Taking secondary transpose on both sides, we have Again taking (A s ) s (A s ) s = I s AA = I ( I s = I) AA = I ( A is hermitian) (3) (A s ) 2 = I A s A s = I Taking secondary transpose on both sides, we have (A s ) s (A s ) s = I s AA = I ( I s = I) ( A is hermitian), ( I = I) (4) Therefore, from Equations (3) and (4), we have AA = A is conjugate unitary. (iii) and (i) (ii) (A s ) 2 = I A s A s = I 79

6 Taking secondary transpose on both sides, we have (A s ) s (A s ) s = I s AA = I ( I s = I) Post multiplying by A on both sides, we have, AAA = IA A (AA ) = A AI = A ( A is conjugate unitary AA = ) A = A A is hermitian Theorem 4. Any two of the following imply the other. (i). A is conjugate unitary matrix (ii). A is secondary unitary matrix (iii). A = A θ Proof. (i) and (ii) (iii) A is conjugate unitary matrix AA = AA = I AA = AA θ Pre multiplying by A 1 on both sides, we have A 1 AA = A 1 AA θ IA = A θ A = A θ Case (ii): ( I = I) A A = A θ A 80

7 Post multiplying by A 1 on both sides, we have A (AA 1 ) = A θ (AA 1 ) A = A θ Therefore, in both cases, we have A = A θ. (ii) and (iii) (i) A is s-unitary AA θ = A θ A = I Case (ii): AA θ = I AA = I A θ A = I ( I = I) Therefore, in both cases, we have AA = A is conjugate unitary matrix. (iii) and (ii) (i) AA = I AA θ = I ( A = A θ ) Case (ii): ( I = I) A θ A = I ( A = A θ ) Therefore, in both cases, we have AA θ = A θ A = I A is s-unitary. 81

8 Definition 5. Two matrices A, B C n n are unitarily similar if there exists a unitary matrix U C n n such that A = U BU. Theorem 6. A is conjugate unitary if every matrix unitarily similar to A, is conjugate unitary. Proof. Assume that A is conjugate unitary AA =. Let B is any matrix, which is unitarily similar to A. Therefore B = U AU where U is any unitary matrix. BB = (U AU) (U AU) Case (ii): = U A (UU ) A U = U AA U ( U is unitary matrix) = U U ( A is conjugate unitary matrix) = I ( U is unitary matrix) B B = (U AU) (U AU) = U A (UU ) AU = U A AU ( U is unitary matrix) = U U ( A is conjugate unitary matrix) = I ( U is unitary matrix) B B = I B B = I ( I = I) Therefore, in both cases, we have BB conjugate unitary. = B B = I B is Theorem 7. Let A C n n. If A is unitarily similar to a diagonal matrix which is conjugate unitary then A is conjugate unitary. Proof. Given that A is unitarily similar to a diagonal matrix D, which is conjugate unitary. Then there exists a unitary matrix 82

9 U such that A = U DU. We have to prove A is conjugate unitary matrix. Case (ii): AA = (U DU) (U DU) = U D (UU ) D U = U DD U( U is unitary matrix) = U U( D is conjugate unitary matrix) = I( U is unitary matrix) A A = (U DU) (U DU) = U D (UU ) DU = U D D( U is unitary matrix) = U U( D is conjugate unitary matrix) = I( U is unitary matrix) ( I = I) Therefore, in both cases, we have AA = A is conjugate unitary matrix. Theorem 8. Let A, B C n n be conjugate unitary matrices. If A is unitarily similar to B then AB is Conjugate unitary matrix. Proof. Given that A, B C n n be conjugate unitary matrices. Assume that A is unitarily similar to B A = U BU B = UAU 83

10 (AB) (AB) = ((U BU) (UAU )) ((U BU) (UAU )) Case (ii): = ((U BU) (UAU )) ((UAU ) ((U BU)) ) = ((U BU) (UAU )) ((UA U ) (U B U)) = U BUUA(U U)A U U B U = U BUU(AA )U U B U( U is unitary matrix) = U BU(UU )U B U( A is conjugate unitary matrix) = U B(UU )B U( U is unitary matrix) = U (BB )U( U is unitary matrix) = U U( B is conjugate unitary matrix) = I( U is unitary matrix) (AB) (AB)= ((U BU) (UAU )) ((U BU) (UAU )) = ((UAU ) ((U BU)) ) ((U BU) (UAU )) = ((UA U ) (U B U)) ((U BU) (UAU )) = UA U U B (UU )BUUAU = UA U U (B B)UUAU ( U is unitary matrix) = UA U (U U)UAU ( B is conjugate unitary matrix) = UA (U U)AU ( U is unitary matrix) = U(A A)U ( U is unitary matrix) = UU ( A is conjugate unitary matrix) = I( A is conjugate unitary matrix) (AB) (AB) = I (AB) (AB) = I ( I = I) Therefore, in both cases, we have (AB) (AB) = (AB) (AB) = I AB is conjugate unitary matrix. Theorem 9. Let A and B be unitarily similar. Then A is conjugate unitary if and only if B is conjugate unitary. 84

11 Proof. Assume that A is conjugate unitary matrix. That is AA =. Since A and B are unitarily similar then there exists a unitary matrix U such that A = U BU B = UAU BB = (UAU ) (UAU ) Case (ii): = UA(U U)A U = UAA U ( U is unitary matrix) = UU ( A is conjugate unitary matrix) = I ( U is unitary matrix) B B = (UAU ) (UAU ) = UA (U U)AU = UA AU ( U is unitary matrix) = UU ( A is conjugate unitary matrix) = I ( U is unitary matrix) B B = I B B = I ( I = I) Therefore, in both cases, we have BB = B B = I B is conjugate unitary matrix. Conversely, assume that B is conjugate unitary matrix. We have to prove A is conjugate unitary matrix. That is to prove AA =. Since A and B are unitarily similar then there exists a unitary matrix U such that A = U BU. AA = (U BU) (U BU) = U B(UU )B U = U BB U ( U is unitary matrix) = U U ( B is conjugate unitary matrix) = I ( U is unitary matrix) 85

12 Case (ii): A A = (U BU) (U BU) = U B (UU ) BU = U B BU ( U is unitary matrix) = U U ( B is conjugate unitary matrix) = I ( U is unitary matrix) ( I = I) Therefore, in both cases, we have AA = A is conjugate unitary matrix. Definition 10. Let A, B C n n. A is said to be s-unitarily similar to B if there exists a s- unitary matrix U such that A = U θ BU. Theorem 11. Let A, B C n n be conjugate unitary matrices. If A is secondary unitarily similar to B then A θ is secondary unitarily similar to B θ. Proof. Given A, B are conjugate unitary matrices. By definition, A is s-unitary similar to B A = U θ BU. Taking conjugate secondary transpose on both sides, we have (A) θ = ( U θ BU ) θ = U θ B θ( U θ) θ = U θ B θ U Therefore A θ is s-unitarily similar to B θ. 3 Conclusion Some equivalent conditions are related to conjugate unitary matrix, hermitian matrix, secondary unitary matrix, secondary transpose of a matrix are obtained. Also some theorems on unitarily similar and s-unitarily similar between two conjugate unitary matrices are derived. 86

13 References [1] Anna Lee, secondary symmetric, secondary skew symmetric, secondary orthogonal matrices, Period. Math. Hungary., 7(1976), [2] Anna Lee, On s-symmetric, s-skew symmetric and s-orthogonal matrices, Period. Math. Hungary., 7(1976), [3] A.Faßbender and Kh.D.Ikramov, Conjugate-normal matrices: a survey, Lin. Alg. Appl., 429(2008), [4] A.Govindarasu and S.Sassicala, Characterizations of conjugate unitary matrices, Journal of ultra scientist of physical science, 29(1)(2017), [5] S.Krishnamoorthy and A.Govindarasu, On Secondary Unitary Matrices, International Journal of Computational Science and Mathematics, 2(3)(2010), [6] S.Krishnamoorthy and R.Vijayakumar, Some equivalent conditions on s-normal matrices, International Journal of Comtemporary Mathematical Sciences, 4(29)(2009), [7] A.K.Sharma, Text book of matrix theory, DPH Mathematics Series, (2004). [8] J.Vermeer, Orthogonal similarity of a real matrix and its transpose, Linear Algebra and its Applications, 428(2008),

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