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
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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
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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)
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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)
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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
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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
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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. ⇒
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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
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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. ⇒
Theorem 8. Let A, B Cn n be conjugate unitary matrices. × If A is unitarily similar to B∈then AB is Conjugate unitary matrix.
Proof. Given that A, B Cn n be conjugate unitary matrices. ∈ × Assume that A is unitarily similar to B A = U ∗BU B = ⇒ ⇒ UAU ∗
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Case (i):
(AB) (AB)∗ = ((U ∗BU)(UAU ∗)) ((U ∗BU)(UAU ∗))∗
= ((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) Case (ii):
(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) Taking conjugate on both sides, we have
(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.
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Proof. Assume that A is conjugate unitary matrix. That is AA∗ = A∗A = I. Since A and B are unitarily similar then there exists a unitary matrix U such that A = U ∗BU B = UAU ∗ Case (i): ⇒
BB∗ = (UAU ∗)(UAU ∗)∗
= UA(U ∗U)A∗U ∗ = UAA∗U ∗ (∵ U is unitary matrix) = UU ∗ (∵ A is conjugate unitary matrix) = I (∵ U is unitary matrix) Case (ii):
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) 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 matrix. ⇒ Conversely, assume that B is conjugate unitary matrix. We have to prove A is conjugate unitary matrix. That is to prove AA∗ = A∗A = I. Since A and B are unitarily similar then there exists a unitary matrix U such that A = U ∗BU. Case (i):
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)
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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) 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. ⇒
Definition 10. Let A, B Cn n. A is said to be s-unitarily × similar to B if there exists a s-∈ unitary matrix U such that A = U θBU.
Theorem 11. Let A, B Cn n be conjugate unitary matrices. × If A is secondary unitarily similar∈ to B then Aθ is secondary unitarily similar to Bθ.
Proof. Given A, B are conjugate unitary matrices. By definition, A is s-unitary similar to B A = U θBU. Taking conjugate secondary transpose on both sides,⇒ we have θ (A)θ = U θBU θ = U θBθ Uθ θ θ = U B U Therefore Aθ is s-unitarily similar to Bθ.
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.
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References
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[3] A.Faßbender and Kh.D.Ikramov, Conjugate-normal matrices: a survey, Lin. Alg. Appl., 429(2008), 1425-1441.
[4] A.Govindarasu and S.Sassicala, Characterizations of conjugate unitary matrices, Journal of ultra scientist of physical science, 29(1)(2017), 33-39.
[5] S.Krishnamoorthy and A.Govindarasu, On Secondary Unitary Matrices, International Journal of Computational Science and Mathematics, 2(3)(2010), 247-253.
[6] S.Krishnamoorthy and R.Vijayakumar, Some equivalent conditions on s-normal matrices, International Journal of Comtemporary Mathematical Sciences, 4(29)(2009), 1449-1454.
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