The η-anti-Hermitian solution to some classic matrix equations

Applied Mathematics and Computation 320 (2018) 264–270

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The η-anti-Hermitian solution to some classic matrix equationsR Xin Liu∗ Faculty of Information Technology, Macao University of Science and Technology, Avenida Wai Long, TaiPa, Macau 999078, PR China

a r t i c l e

i n f o

a b s t r a c t We in this paper consider the η-anti-Hermitian solution to some classic matrix equations AX = B, AXB = C, AXAη∗ = B, E XE η∗ + F Y F η∗ = H, respectively. We derive the necessary and sufficient conditions for the above matrix equations to have η-anti-Hermitian solutions and also provide the general expressions of solutions when those equations are solvable. As applications, for instance, we give the solvability conditions and general η-anti-Hermitian solution to equation system AX = B, CY = D, MXMη∗ + NY N η∗ = G.

Keywords: Quaternion matrix equation η-anti-Hermitian matrix Moore–Penrose inverse

© 2017 Elsevier Inc. All rights reserved.

1. Introduction Throughout this paper, we denote the set of all m × n matrices over the quaternion division algebra

H = {a0 + a1 i + a2 j + a3 k|i2 = j2 = k2 = i jk = −1, a0 , a1 , a2 , a3 ∈ R} by Hm×n ; the symbols I, AT , A∗ , r(A) stand for the identity matrix with the appropriate size, the transpose, the conjugate transpose and the rank of a matrix A over H, respectively. Moreover, the Moore–Penrose inverse of A ∈ Hm×n , denoted by A† , is a unique solution to the matrix equation system ∗

∗

AX A = A, X AX = X, (AX ) = AX, (X A ) = X A. And LA = I − A† A and RA = I − AA† are two projectors induced by A. Thus, ∗

LA = ( LA ) = ( LA ) , ∗

2

RA = ( RA ) = ( RA ) . 2

Quaternion matrices have wide applications in signal and color image processing, computer science, quantum physics, robotics, statistical signal processing. For instance, η-Hermitian matrix has important applications in linear modeling and convergence analysis in statistical signal processing (see [11–13]). We in the paper will consider those kinds of matrices over H. Now, we first introduce the definition. Definition 1.1 ([3,9,10]). A matrix A ∈ Hn×n is η-Hermitian if A = Aη∗ , where Aη∗ = −ηA∗ η, η ∈ {i, j, k}. A matrix A ∈ Hn×n is η-anti-Hermitian if A = −Aη∗ , where Aη∗ = −ηA∗ η, η ∈ {i, j, k}. R ∗

This research was supported by Macao Science and Technology Development Fund (No. 003/2015/A1) Corresponding author. E-mail address: [email protected]

https://doi.org/10.1016/j.amc.2017.09.033 0 096-30 03/© 2017 Elsevier Inc. All rights reserved.

X. Liu / Applied Mathematics and Computation 320 (2018) 264–270

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And the following are some basic properties of η-Hermitian matrix and η-anti-Hermitian matrix. Lemma 1.1 ([8]). Let A be the matrix over H. Then (a) (b) (c) (d) (e) (f)

(A + B )η∗ = Aη∗ + Bη∗ . (AB )η∗ = Bη∗ Aη∗ . (Aη∗ )η∗ = A. η∗ (Aη∗ )† = (A† ) . η ∗ r ( A ) = r ( A ) = r ( Aη ) = r ( Aη Aη ∗ ) = r ( Aη ∗ Aη ). (RA )η∗ = LAη∗ , (LA )η∗ = RAη∗ .

η-(anti)-Hermicity is generalized concept over H. By Definition 1.1, A is j-(anti)-Hermitian and also k-(anti)-Hermitian over H, if A is a complex (skew-)symmetric matrix, i.e., A = ±AT . A is i-(anti)-Hermitian if A is a complex (skew-)hermitian matrix, i.e., A = ±A∗ . As a kind of matrices which including so many useful and important matrices, it’s really necessary to be investigated. Recently, η-Hermitian matrix was widely investigated. Horn and Zhang [9] gave a singular value decomposition for η-Hermitian matrices. He and Wang [8] derived the solvability conditions of the quaternion matrix equation η∗ AX + (AX ) + BY Bη∗ + C ZC η∗ = D. Further, they gave the general η-Hermitian solution by Moore–Penrose inverse. Yuan et al. [3–5] by using Kronecker product and complex representation of quaternion matrix, presented the least square ηHermitian/ η-anti-Hermitian solution to some quaternion matrix equations. Such as AXB + CXD = E and (AXB, CXD ) = (E, F ). Belk and Ahmadi-Asl [6] presented an iterative method for determining η-Hermitian and η-anti-Hermitian least-squares solutions to the quaternion matrix equation AXB + CY D = E. For more related work, one may refer to [1,2,7]. To our knowledge, η-anti-Hermitian solution to following classic equations AX = B AX B = C AX Aη∗ = B E X E η∗ + F Y F η∗ = H has not been considered. In this paper, we derive the solvability conditions and general η-anti-Hermitian solutions to above equations, respectively. As applications, we can solve the η-anti-Hermicity problem to the following three different equation systems:

AX = B, XC = D. AX = B, E X E η∗ = G. AX = B, CY = D, MX Mη∗ + NY N η∗ = G. 2. η-anti-Hermitian solutions to some classic equations In this section, we consider the η-anti-Hermitian solutions to simple but classic matrix equations. Now, we start with the matrix equation

AX = B.

(2.1)

Theorem 2.1. Let A, B be the matrices over H with appropriate sizes. Then (a) Matrix Eq. (2.1) has an η-anti-Hermitian solution if and only if RA B = 0, ABη∗ = −BAη∗ . (b) If conditions in (a) hold, then the general η-anti-Hermitian solution is given by η∗

X = A† B − ( A† B )

η∗

+ A† (ABη∗ )(A† )

+ LA U LA η ∗ ,

(2.2)

where U = −U η∗ is an arbitrary matrix over H with appropriate size. Proof. Assume that (2.1) has an η-anti-Hermitian solution C. Then RA B = RA AC = 0, ABη∗ = AC η∗ Aη∗ = −ACAη∗ = −BAη∗ . η∗ η∗ Conversely, assume that RA B = 0, ABη∗ = −BAη∗ . Then it’s not difficult to verify that A† B − (A† B ) + A† (ABη∗ )(A† ) is a solution to (2.1). Assume X0 is an arbitrary η-anti-Hermitian solution, we now show that each solution X of (2.1) is in the form of (2.2). Setting U = X0 gives η∗

X = A† B − ( A† B )

η∗

+ A† (ABη∗ )(A† )

+ LA X0 LA η∗

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X. Liu / Applied Mathematics and Computation 320 (2018) 264–270

η∗

= A† B − ( A† B )

η∗

= X0 − Bη∗ (A† )

η∗

+ A† (ABη∗ )(A† ) η∗

η∗

+ (X0 − A† B )(I − (A† A ) )

− X0 (A† A ) η∗

= X0 − (A† B + A† A(−X0 ))

= X0 ,

which implies that each η-anti-Hermitian solution to (2.1) can be expressed as (2.2).



Next, we consider another classic matrix equation

AX Aη∗ = B, B = −Bη∗ .

(2.3)

We first begin with a well-known result about AXB = C. Lemma 2.2 ([14] and [15]). Let A, B, C be matrices over H with appropriate sizes. Then AXB = C is consistent if and only if RAC = 0, CLB = 0. And in this case, the general solution is given by X = A†CB† + LAU + W RB , where U, W are arbitrary matrices over H with appropriate sizes. Theorem 2.3. Let A, B be the matrices over H with appropriate sizes. Then (a) Matrix Eq. (2.3) has an η-anti-Hermitian solution if and only if RA B = 0. (b) If condition in (a) hold, then the general solution is given by

X = A† B ( Aη ∗ ) + LA V − V η ∗ LA η ∗ , †

where V is an arbitrary matrix over H with appropriate size. Proof. Note that (2.3) has an η-anti-Hermitian solution X if and only if the matrix system

AY Aη∗ = B, A(−Y η∗ )Aη∗ = B

(2.4)

has a solution Y, where Y don’t have to be an η-anti-Hermitian matrix. If (2.4) is consistent, then

X=

1 (Y − Y η∗ ). 2

Since equation system (2.4) is equivalent to the matrix equation

AY Aη∗ = B, B = −Bη∗ .

(2.5)

Hence, if (2.5) is consistent for Y, then (2.3) is also consistent for an η-anti-Hermitian X. Then by Lemma 2.2, (2.5) is consistent if and only if

RA B = 0, Bη ∗ LAη ∗ = 0, η∗ Since Bη∗ LAη∗ = Bη∗ RA η∗ = (RA B ) . Therefore, the solvability conditions for (2.5) can be reduced to RA B = 0. In this case

Y = A† B ( Aη ∗ ) + LA U + W RAη ∗ . †

Hence

X= where V =

1 (Y − Y η∗ ) = A† B(Aη∗ )† + LAV − V η∗ LA η∗ , 2 1 2 (U

− W η∗ ), U, W are arbitrary matrices over H with appropriate sizes.

Next, Using the fact that X + X η∗ = 0 if and only if X is



η-anti-Hermitian, we can obtain the η-anti-Hermitian solution

to

AX B = C.

(2.6)

Before giving our statement, we first introduce the following result. Lemma 2.4 ([8]). Let A, B = Bη∗ be the matrices over H with appropriate sizes. Then the equation AX + (AX ) η∗ if and only if RA B(RA ) = 0. In this case, the general solution can be expressed as

X = A† B −

η∗

= B has a solution

1 † η∗ η∗ A B(A† ) Aη∗ + LAV + U η∗ (A† ) Aη∗ − A†UAη∗ , 2

where U, V are arbitrary matrices over H with appropriate sizes. η∗ Theorem 2.5. Let A, B, C be the matrices over H with appropriate sizes, and denote Aˆ = [RB η∗ , LA ], Bˆ = −[A†CB† + (A†CB† ) ]. Then

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(a) The matrix Eq. (2.6) has an η-anti-Hermitian solution if and only if

RAC = 0, CLB = 0, η∗

R(RB η∗ +LA ) [A†CB† + (A†CB† ) ]R(RB η∗ +LA ) = 0. (b) If conditions in (a) hold, then the general solution is given by

X = A†CB† + LA SUˆ + Uˆ η∗ T RB , Uˆ = Aˆ † Bˆ −

η∗ 1 ˆ † ˆ † η∗ ˆ η∗ A Bˆ(A ) A + LAˆ V + U η∗ (Aˆ † ) Aˆ η∗ − Aˆ †U Aˆ η∗ , 2

(2.7)

where S = [0, I], T = [0I ], U, V are arbitrary matrices over H with appropriate sizes. Proof. By Lemma 2.2, (2.6) is consistent if and only if RAC = 0, CLB = 0 and the general solution to AXB = C is given by

X = A†CB† + LAU1 + V1 RB , where U1 , V1 are arbitrary matrices over H with appropriate sizes. X is an η-anti-Hermitian matrix if and only if X + X η∗ = 0, i.e., η∗

A†CB† + LAU1 + V1 RB + (A†CB† + LAU1 + V1 RB )

= 0,

which can be rewritten as follows η∗

AˆUˆ + (AˆUˆ )

= Bˆ,

(2.8)

where

Aˆ = [RB η∗ , LA ], η∗

Bˆ = −[A†CB† + (A†CB† ) ],



Uˆ =

η∗ 

V1

U1

.

η∗ By Lemma 2.4, the Eq. (2.8) has a solution if and only if RAˆ Bˆ(RAˆ ) = 0. η ∗ Since RB = LBη∗ and LA are projectors. Therefore RAˆ = R(R η∗ R +L L η∗ ) = R(R η∗ +L ) . B B B A A A Then the solvability condition for (2.8) is η∗

R(RB η∗ +LA ) [A†CB† + (A†CB† ) ]R(RB η∗ +LA ) = 0. Also by using Lemma 2.4,

Uˆ = Aˆ † Bˆ −

η∗ 1 ˆ † ˆ † η∗ ˆ η∗ A Bˆ(A ) A + LAˆ V + U η∗ (Aˆ † ) Aˆ η∗ − Aˆ †U Aˆ η∗ , 2

(2.9)

where U, V are arbitrary matrices over H with appropriate sizes. Since

U1 = SUˆ , V1 = Uˆ η∗ T , then our η-anti-Hermitian solution is given by

X = A†CB† + LAU1 + V1 RB = A†CB† + LA SUˆ + Uˆ η∗ T RB , with Uˆ in (2.9).



Basing on the results of Theorems 2.3 and 2.5, we now consider the matrix equation

E X E η∗ + F Y F η∗ = H,

(2.10)

with H = −H η∗ . Theorem 2.6. Let E, F, H be the matrices over H with appropriate sizes, and denote

Aˆ = RE F , Bˆ = F η∗ , Cˆ = RE H.  = [Rη ∗ , L ] A Aˆ ˆ B

 = −[Aˆ †CˆBˆ† + (Aˆ †CˆBˆ† )η∗ ]. B Then

(2.11)

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X. Liu / Applied Mathematics and Computation 320 (2018) 264–270

(a) The matrix Eq. (2.10) has an η-anti-Hermitian solution if and only if

RAˆ Cˆ = 0, CˆRF η∗ = 0, η∗

† † R(LF +LAˆ ) [Aˆ †Cˆ(F η∗ ) + (Aˆ †Cˆ(F η∗ ) ) ]R(LF +LAˆ ) = 0.

(b) If conditions in (a) hold, then the general solution is given by

X = E † H ( E η ∗ ) − E † F Y F η ∗ ( E η ∗ ) + LE W − W η ∗ LE η ∗ , †

†

† Y = Aˆ †Cˆ(F η∗ ) + LAˆ SUˆ + Uˆ η∗ T RF η∗ ,

† B − Uˆ = A

(2.12)

1 ˆ †  † η∗ η∗ †U A † )η∗ A η∗ − A η∗ , A B(A ) A + LAV + U η∗ (A 2

where S = [0, I], T =

I 0

, W, U, V are arbitrary matrices over H with appropriate sizes.

Proof. By Theorem 2.3, we obtain that

E X E η∗ + F Y F η∗ = H is consistent for X = −X η∗ if and only if η∗

H − F Y F η ∗ = − (H − F Y F η ∗ ) and

RE ( H − F Y F η ∗ ) = 0.

(2.13)

In this case, the solution is given by

X = E † ( H − F Y F η ∗ ) ( E η ∗ ) + LE W − W η ∗ LE η ∗ . †

Note that the Eq. (2.13) is in the form of

AˆY Bˆ = Cˆ,

(2.14)

where Aˆ , Bˆ, Cˆ are given by (2.11). Thus by Theorem 2.5, the Eq. (2.14) is consistent if and only if

RAˆ Cˆ = 0, CˆRF η∗ = 0, † η∗

R(LF +LAˆ ) [Aˆ †Cˆ(F η∗ ) + (Aˆ †Cˆ(F η∗ ) ) ]R(LF +LAˆ ) = 0. †

In this case, Y is given by (2.12).



3. Applications In this section, we have more about some matrix equation systems by using the results of Section 2. Corollary 3.1. Let A, B, C, D be the matrices over H with appropriate sizes, and denote



Aˆ =







A B , Bˆ = . −C η∗ Dη ∗

Then (a) Matrix equation system

AX = B, XC = D

(3.1)

has an η-anti-Hermitian solution if and only if RAˆ Bˆ = 0, ABη∗ = −BAη∗ , CDη∗ = −DC η∗ , AD = BC. (b) If conditions in (a) hold, then the general η-anti-Hermitian solution is given by η∗

X = Aˆ † Bˆ − (Aˆ † Bˆ )

η∗

+ Aˆ † (Aˆ Bˆη∗ )(Aˆ † )

+ LAˆ U LAˆ η∗ ,

where U = −U η∗ is an arbitrary matrix over H with appropriate size.

X. Liu / Applied Mathematics and Computation 320 (2018) 264–270

269

Proof. Obviously, equation system (3.1) is equivalent to AX = B, C η∗ X η∗ = Dη∗ . Thus, (3.1) has an anti-η-Hermitian solution X if and only if AX = B, C η∗ X = −Dη∗ . Taking



Aˆ =







A B , Bˆ = , −C η∗ Dη ∗

then our statements follow immediately by Theorem 2.1.



Combing the results of Theorem 2.1 and Theorem 2.3, we can solve the η-anti-Hermitian problem of

AX = B, E X E η∗ = G,

(3.2)

with G = −Gη∗ . η∗ Corollary 3.2. Let A, B, E, G be the matrices over H with appropriate sizes, and denote Aˆ = ELA , Bˆ = G − E[A† B − (A† B ) + η∗ η∗ † η ∗ † A (AB )(A ) ]E . Then

(a) Matrix equation system (3.2) has an η-anti-Hermitian solution if and only if RA B = 0, ABη∗ = −BAη∗ , RAˆ Bˆ = 0. (b) If conditions in (a) hold, then the general solution is given by η∗

X = A† B − ( A† B )

η∗

+ A† (ABη∗ )(A† )

η∗

†

+ LA Aˆ † Bˆ(Aˆ η∗ ) LA η∗ + LA LAˆ V LA η∗ − (LA LAˆ V LA η∗ ) ,

where V is an arbitrary matrix over H with appropriate size. By Theorem 2.1 and Theorem 2.6, we have the following results. Corollary 3.3. Let A, B, M, N, G be the matrices over over H with appropriate sizes, and denote

E = MLA , F = NLC , η∗

H = G − M[A† B − (A† B )

η∗

Aˆ = RE F , Bˆ = F η∗ , Cˆ = RE H.  = [Rη ∗ , L ] , B  = −[Aˆ †CˆBˆ† + (Aˆ †CˆBˆ† )η∗ ]. A Aˆ ˆ B

Then (a) The matrix equation system

AX = B, CY = D, MX Mη∗ + NY N η∗ = G with G = −Gη∗ has an η-anti-Hermitian solution if and only if

RA B = 0, ABη∗ = −BAη∗ , RC D = 0, CDη∗ = −DC η∗ , RAˆ Cˆ = 0, CˆRF η∗ = 0, † η∗

R(LF +LAˆ ) [Aˆ †Cˆ(F η∗ ) + (Aˆ †Cˆ(F η∗ ) ) ]R(LF +LAˆ ) = 0. †

(b) If the conditions in (a) hold, then the general solution is given by η∗

+ A† (ABη∗ )(A† )

η∗

+ C † (CDη∗ )(C † )

X = A† B − ( A† B )

Y = C † D − (C † D )

η∗

+ LAU1 LA η∗ ,

η∗

+ LC U2 LC η∗ ,

U1 = E † H (E η∗ ) − E † F U2 F η∗ (E η∗ ) + LE W − W η∗ LE η∗ , †

η∗

+ A† (ABη∗ )(A† ) ]Mη∗ − N[C † D − (C † D )

†

η∗

+ C † (CDη∗ )(C † ) ]N η∗ .

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X. Liu / Applied Mathematics and Computation 320 (2018) 264–270

† U2 = Aˆ †Cˆ(F η∗ ) + LAˆ SUˆ + Uˆ η∗ T RF η∗ ,

† B − Uˆ = A

1 ˆ †  † η∗ η∗ †U A † )η∗ A η∗ − A η∗ , A B(A ) A + LAV + U η∗ (A 2

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