International Journal of Pure and Applied Mathematics Volume 119 No. 6 2018, 75-88 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue 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. ∗[email protected] 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 AT and secondary transpose As are related as As = VAT V , where ‘V ’ is the permutation matrix with units in s its secondary diagonal. Also A denotes the conjugate secondary s transpose of A. i.e., (Ac = ij) where cij = an j+1,n i+1 [2]. − − Any square matrix over a field is similar to its transpose and any 75 International Journal of Pure and Applied Mathematics Special Issue 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 Cn n be the space of n n complex matrices of order n. For × T × s s θ A Cn n. Let A , A, A∗, A , A (= A ) denote transpose, × conjugate,∈ conjugate transpose, secondary transpose, conjugate secondary transpose of a matrix A respectively. Also the T 2 permutation matrix V satisfies V = V = V ∗ = V and V = I. A matrix A Cn n is called unitary if AA∗ = A∗A = I [7] • ∈ × A matrix A Cn n is called conjugate normal if AA∗ = A∗A × • [3] ∈ A matrix A Cn n is called s-normal if and only if × • AAθ = AθA [6]∈ A matrix A Cn n is called secondary unitary (s-unitary) if × • AAθ = AθA ∈= I [5] θ s A matrix A Cn n is called s-hermitian if A = A = A • ∈ × A matrix A Cn n is called conjugate unitary if • ∈ × AA∗ = A∗A = I [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∗ = A∗A = I s s To show A (A )∗ = (As)∗As = I Case (i): AA∗ = I 76 International Journal of Pure and Applied Mathematics Special Issue Taking secondary transpose on both sides, we have s s (AA∗) = I s s s (A∗) A = I (∵ I = I) s s (A )∗A = I Taking conjugate on both sides, we have (As)∗As = I (As)∗As = I (∵ I = I) (1) Case (ii): A∗A = I Taking conjugate on both sides, we have A∗A = I A∗A = I (∵ I = I) Taking secondary transpose on both sides we have s s (A∗A) = I s s s A (A∗) = I (∵ I = I) s s A (A )∗ = I (2) s s Therefore, from equations (1) and (2), we have A (A )∗ = (As)∗As = I Secondary transpose of A is conjugate unitary matrix. ⇒ Theorem 2. If A is conjugate unitary matrix then VA is conjugate unitary matrix. Proof. A is conjugate unitary matrix AA∗ = A∗A = I Case (i): ⇒ (VA)(VA)∗ = V AA∗V ∗ = V AA∗V (∵ V ∗ = V ) = VV (∵ A is conjugate unitary) = V 2 2 = I (∵ V = I) 77 International Journal of Pure and Applied Mathematics Special Issue Case (ii): (V A)∗ (VA) = A∗V ∗VA = A∗VVA (∵ V ∗ = V ) 2 = A∗V A 2 = A∗A (∵ V = I) = I (∵ A is conjugate unitary) Taking conjugate on both sides, we have (AV) ∗ (VA) = I (AV) ∗ (VA) = I (∵ I = I) Therefore, in both cases, we have (VA)(VA)∗ = (AV) ∗ (VA) = I VA is conjugate unitary matrix. ⇒ Theorem 3. Any two of the following imply the other. (i). A is conjugate unitary (ii). A is hermitian (iii). Asis involutary Proof. (i) and (ii) (iii) ⇒ A is conjugate unitary AA∗ = A A = I ⇒ ∗ Case (i): AA∗ = I Taking secondary transpose on both sides, we have s s (AA∗) = I s s s (A∗) A = I (∵ I = I) s s A A = I (∵ A is hermitian) (As)2 = I Case (ii): A∗A = I Taking conjugate transpose on both sides, we have A∗A = I A∗A = I (∵ I = I) 78 International Journal of Pure and Applied Mathematics Special Issue Taking secondary transpose on both sides, we have s s (A∗A) = I s s s A (A∗) = I (∵ I = I) s s A A = I (∵ A is hermitian) (As)2 = I Therefore, in both cases, we have (As)2 = I As is involutary. (ii) and (iii) (i) ⇒ ⇒ (As)2 = I AsAs = I Taking secondary transpose on both sides, we have (As)s(As)s = Is s AA = I (∵ I = I) AA∗ = I (∵ A is hermitian) (3) Again taking (As)2 = I AsAs = I Taking secondary transpose on both sides, we have (As)s(As)s = Is s AA = I (∵ I = I) A∗A = I (∵ A is hermitian) Taking conjugate on both sides, we have, A∗A = I A∗A = I (∵ I = I) (4) Therefore, from Equations (3) and (4), we have AA∗ = A∗A = I A is conjugate unitary. ⇒ (iii) and (i) (ii) ⇒ (As)2 = I AsAs = I 79 International Journal of Pure and Applied Mathematics Special Issue Taking secondary transpose on both sides, we have (As)s(As)s = Is s AA = I (∵ I = I) Post multiplying by A∗ on both sides, we have, AAA∗ = IA∗ A (AA∗) = A∗ AI = A∗ ( A is conjugate unitary AA∗ = A A = I) ∵ ⇒ ∗ 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 θ (iii). A∗ = A Proof. (i) and (ii) (iii) ⇒ A is conjugate unitary matrix AA∗ = A∗A = I Case (i): ⇒ AA∗ = I θ AA∗ = AA 1 Pre multiplying by A− on both sides, we have 1 1 θ A− AA∗ = A− AA θ IA∗ = A θ A∗ = A Case (ii): A∗A = I Taking conjugate on both sides, we have A∗A = I A∗A = I (∵ I = I) θ A∗A = A A 80 International Journal of Pure and Applied Mathematics Special Issue 1 Post multiplying by A− on both sides, we have 1 θ 1 A∗(AA− ) = A (AA− ) θ A∗ = A θ Therefore, in both cases, we have A∗ = A . (ii) and (iii) (i) A is s-unitary⇒ AAθ = AθA = I Case (i): ⇒ AAθ = I AA∗ = I Case (ii): AθA = I A∗A = I Taking conjugate on both sides, we have A∗A = I A∗A = I (∵ I = I) Therefore, in both cases, we have AA∗ = A∗A = I A is conjugate unitary matrix. ⇒ (iii) and (ii) (i) Case (i): ⇒ AA∗ = I θ θ AA = I (∵ A∗ = A ) Case (ii): A∗A = I Taking conjugate on both sides, we have A∗A = I A∗A = I (∵ I = I) θ θ A A = I (∵ A∗ = A ) Therefore, in both cases, we have AAθ = AθA = I A is s-unitary. ⇒ 81 International Journal of Pure and Applied Mathematics Special Issue Definition 5. Two matrices A, B Cn n are unitarily similar ∈ × if there exists a unitary matrix U Cn 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∗ = A∗A = I. Let B is any matrix, which is unitarily similar⇒ to A. Therefore B = U ∗AU where U is any unitary matrix. Case (i): BB∗ = (U ∗AU)(U ∗AU)∗ = U ∗A (UU ∗) A∗U = U ∗AA∗U (∵ U is unitary matrix) = U ∗U (∵ A is conjugate unitary matrix) = I (∵ U is unitary matrix) Case (ii): 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) Taking conjugate on both sides, we have B∗B = I B∗B = I (∵ I = I) Therefore, in both cases, we have BB∗ = B∗B = I B is conjugate unitary. ⇒ Theorem 7. Let A Cn 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 International Journal of Pure and Applied Mathematics Special Issue U such that A = U ∗DU. We have to prove A is conjugate unitary matrix. Case (i): 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) Case (ii): 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) Taking conjugate on both sides, we have A∗A = I A∗A = I (∵ I = I) Therefore, in both cases, we have AA∗ = A∗A = I A is conjugate unitary matrix.