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A system of generalized Sylvester quaternion matrix equations and its applications
Applied Mathematics and Computation 273 (2016) 74–81
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A system of generalized Sylvester quaternion matrix equations and its applications Xiang Zhang∗ Department of Mathematics, Guizhou Normal University, Guiyang 550001, PR China
a r t i c l e
i n f o
MSC: 15A03 15A09 15A24 11R52 15B57 Keywords: System of matrix equation Quaternion General solution η-Hermitian Moore–Penrose inverse
a b s t r a c t Let Hm×n be the set of all m × n matrices over the real quaternion algebra. We call that A ∈ Hn×n is η-Hermitian if A = −ηA∗ η, η ∈ {i, j, k}, where i, j, k are the quaternion units. Denote Aη∗ = −ηA∗ η. In this paper, we derive some necessary and sufficient conditions for the solvability to the system of generalized Sylvester real quaternion matrix equations Ai Xi + Yi Bi + Ci ZDi = Ei , (i = 1, 2), and give an expression of the general solution to the above mentioned system. As applications, we give some solvability conditions and general solution for the generalized Sylvester real quaternion matrix equation A1 X + Y B1 + C1 ZD1 = E1 , where Z is required to be η-Hermitian. We also present some solvability conditions and general solution for the sysη∗ tem of real quaternion matrix equations involving η-Hermicity Ai Xi + (Ai Xi )η∗ + BiY Bi = Ci , (i = 1, 2), where Y is required to be η-Hermitian. Our results include some well-known results as special cases. © 2015 Elsevier Inc. All rights reserved.
1. Introduction Throughout the paper, the set of all m × n matrices over the quaternion number field H
H = {a0 + a1 i + a2 j + a3 k | i2 = j2 = k2 = i jk = −1, a0 , a1 , a2 , a3 ∈ R} is denoted by Hm×n . For a matrix A, A∗ stands for the conjugate transpose of A. I denotes the identity matrix with appropriate sizes. The Moore–Penrose inverse A† of A, is defined to be the unique matrix A† , such that
(i) AA† A = A, (ii) A† AA† = A† , (iii) (AA† )∗ = AA† , (iv) (A† A)∗ = A† A. Furthermore, LA and RA stand for the two projectors LA = I − A† A and RA = I − AA† induced by A, respectively. It is known that LA = L∗A and RA = R∗A . Quaternions were introduced by Irish mathematician Sir William Rowan Hamilton in 1843. Quaternions are associative and noncommutative. Nowadays quaternion matrices have been widely and heavily used in computer science, quantum physics, signal and color image processing, and so on (e.g. [4,21,22,24,25,38]). Many problems can be reduced to solving systems of quaternion matrix equations. Many problems in systems and control theory require the solution of Sylvester matrix equation. The Sylvester matrix equation has a huge amount of practical applications in feedback control ([5,18,32,36,37]), robust control [20], pole/eigenstructure assignment design [7], neural network [34], and so on. ∗
Corresponding author. Tel.: +14785520175. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.amc.2015.09.074 0096-3003/© 2015 Elsevier Inc. All rights reserved.
X. Zhang / Applied Mathematics and Computation 273 (2016) 74–81
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Roth [23] gave a necessary and sufficient condition for the solvability to the generalized Sylvester matrix equation
A1 X + Y B1 = C1 .
(1.1)
Bakasalary and Kala [3] derived the general solution to the generalized Sylvester matrix equation (1.1) by using generalized inverses. Bai [2] considerd iterative solutions of the continuous Sylvester equations of the form AX + XB = C. Li et al. [8] established a generalized HSS (GHSS) iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and positive definite/semidefinite matrices. Lin and Wei [31] investigated the condition numbers of the generalized Sylvester equation of the form
AX − Y B = C, DX − Y E = F.
In recent years, the study on mixed generalized Sylvester matrix equations are active ([1,6,9–12,29,30,39]). Lee and Vu [19] considered the generalized mixed Sylvester matrix equations
A1 X1 + ZB1 = C1 ,
(1.2)
A2 X2 + ZB2 = C2 ,
where Ai , Bi and Ci are given complex matrices. They gave a necessary and sufficient condition for the solvability to the generalized mixed Sylvester matrix equations (1.2). Wang and He [27] presented necessary and sufficient solvability conditions for the system (1.2), and gave general solution when it is solvable. Recently, He and Wang [14] considered the other form of the generalized mixed Sylvester matrix equations, i.e.,
A1 X1 + ZB1 = C1 ,
(1.3)
A2 Z + X2 B2 = C2 ,
where Ai , Bi and Ci are given complex matrices. He and Wang presented some necessary and sufficient solvability conditions for the system (1.3) and derived an expression of the general solution to (1.3). Motivated by the work mentioned above and the recent increasing interests in quaternion matrices and generalized mixed Sylvester matrix equations, we consider the following more general system of generalized Sylvester real quaternion matrix equations
A1 X1 + Y1 B1 + C1 ZD1 = E1 ,
(1.4)
A2 X2 + Y2 B2 + C2 ZD2 = E2 ,
where Ai , Bi , Ci , Di and Ei (i = 1, 2) are given real quaternion matrices. The generalized Sylvester matrix equation (1.1), mixed Sylvester matrix equations (1.2) and (1.3) are special cases of the system (1.4). One contribution of this paper is to give some solvability conditions and general solution to the system of generalized Sylvester real quaternion matrix equations (1.4). Recall that a quaternion matrix A is called an η-Hermitian matrix if A = Aη∗ = −ηA∗ η, and η ∈ {i, j, k} [26]. The η-Hermitian matrices have some applications widely used in linear modelling, and convergence analysis in statistical signal processing [25]. Horn and Zhang [17] presented an analogous special singular value decomposition for η-Hermitian matrices. Yuan and Wang [33] gave the expressions of the least squares η-Hermitian solution of the real quaternion matrix equation AXB + CXD = E. He and Wang [16] derived some solvability conditions and general solution to the real quaternion matrix equation involving η-Hermicity, i.e., η∗
η∗
A1 X + (A1 X )η∗ + B1Y B1 + C1 ZC1 = D1 , where Y and Z are required to be η-Hermitian. They investigated some properties of the quaternion matrix Aη∗ . Zhang and Wang [35] provided some solvability conditions and the general solution to a system of quaternion matrix equations involving η-Hermicity:
A1 X = C1 , A2Y = C2 , ZB2 = D2 , η∗
η∗
A3 X + (A3 X )η∗ + B3Y B3 + C3 ZC3 = D3 , where Y = Y η∗ , Z = Z η∗ . Recently, He and Wang [13] considered the η-Hermitian solution to a system of real quaternion matrix equations
⎧ ⎨A1 X = C1 , XB1 = D1 , A2Y = C2 , Y B2 = D2 ,
⎩
η∗
η∗
C3 XC3 + D3Y D3 = A3 .
As applications of the system (1.4), we can consider the following two real quaternion matrix equations involving η-Hermicity
A1 X + Y B1 + C1 ZD1 = E1 , and
Z = Z η∗
(1.5)
η∗
A1 X + (A1 X )η∗ + B1Y B1 = C1 , η∗ A2 Z + (A2 Z )η∗ + B2Y B2 = C2 ,
(1.6)
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X. Zhang / Applied Mathematics and Computation 273 (2016) 74–81
where Y = Y η∗ . The other goal of this paper is to give some solvability conditions and general solution to the real quaternion matrix equations (1.5) and (1.6), respectively. The remainder of the paper is organized as follows. In Section 2, we establish some necessary and sufficient conditions for the existence of a solution to the system of generalized Sylvester real quaternion matrix equations (1.4) and derive the explicit expression of the general solution to system (1.4). In Sections 3 and 4, as applications, we give some solvability conditions and the general solutions to real quaternion matrix equations involving η-Hermicity, i.e., the generalized Sylvester real quaternion matrix equation (1.5) and the system of real quaternion matrix equations (1.6), respectively. 2. Some solvability conditions and the general solution to the system (1.4) In this section, using the inverses of the coefficient matrices, we give some solvability conditions for the system of generalized Sylvester real quaternion matrix equations (1.4) to possess a solution and to present an expression of this general solution when it is solvable. We begin with the following lemma. Lemma 2.1 ([15,28]). Let A1 ∈ Hm×n1 , B1 ∈ H p1 ×q , C3 ∈ Hm×n2 , D3 ∈ H p2 ×q , C4 ∈ Hm×n3 , D4 ∈ H p3 ×q , and E1 ∈ Hm×q be given. Set
A = RA1 C3 , B = D3 LB1 , C = RA1 C4 , D = D4 LB1 , E = RA1 E1 LB1 , M = RAC, N = DLB , S = CLM . Then the equation
A1 X1 + X2 B1 + C3 X3 D3 + C4 X4 D4 = E1
(2.1)
is consistent if and only if
RM RA E = 0,
ELB LN = 0,
RA ELD = 0,
RC ELB = 0.
(2.2)
In this case, the general solution of (2.1) can be expressed as
X1 = A†1 (E1 − C3 X3 D3 − C4 X4 D4 ) − A†1W2 B1 + LA1 W1 , X2 = RA1 (E1 − C3 X3 D3 − C4 X4 D4 )B†1 + A1 A†1W2 + W3 RB1 , X3 = A† EB† − A†CM† EB† − A† SC † EN† DB† − A† SV4 RN DB† + LAV3 + V4 RB , X4 = M† ED† + S† SC † EN† + LM LSU1 + LMV4 RN + V5 RD , where V1 , V2 , V3 , V4 , V5 , W1 , W2 , W3 are arbitrary matrices over H with appropriate sizes. Now we give the fundamental theorem of this paper. Theorem 2.1. Let Ai , Bi , Ci , Di and Ei (i = 1, 2) be given. Set
A3 = RA1 C1 , B3 = D1 LB1 , C3 = RA1 E1 LB1 , A4 = C2 LA3 , B4 = RB3 D2 , C4 = E2 − C2 A†3C3 B†3 D2 , A = RA2 A4 , B = D2 LB2 , C = RA2 C2 , D = B4 LB2 , E = RA2 C4 LB2 , M = RAC, N = DLB , S = CLM . Then the system of generalized Sylvester real quaternion matrix equations (1.4) is consistent if and only if
RA3 C3 = 0, C3 LB3 = 0, RM RA E = 0, ELB LN = 0, RA ELD = 0, RC ELB = 0.
(2.3)
In this case, the general solution of (1.4) can be expressed as
Z = A†3C3 B†3 + LA3 U + V RB3 , X1 = A†1 (E1 − C1 ZD1 ) − A†1W1 B1 + LA1 W2 , Y1 = RA1 (E1 − C1 ZD1 )B†1 + A1 A†1W1 + W3 RB1 , X2 = A†2 (C4 − A4UD2 − C2V B4 ) − A†2W4 B2 + LA2 W5 ,
(2.4)
Y2 = RA2 (C4 −
(2.5)
)
A4UD2 − C2V B4 B†2
+
A2 A†2W4
+ W6 RB2 ,
where
U = A† EB† − A†CM† EB† − A† SC † EN† DB† − A† SW7 RN DB† + LAW8 + W9 RB ,
(2.6)
V = M† ED† + S† SC † EN† + LM LSW10 + LMW7 RN + W11 RD ,
(2.7)
and W1 , . . . , W11 are arbitrary matrices over H with appropriate sizes.
X. Zhang / Applied Mathematics and Computation 273 (2016) 74–81
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Proof. We separate the equations in system (1.4) into two groups
A1 X1 + Y1 B1 + C1 ZD1 = E1 ,
(2.8)
A2 X2 + Y2 B2 + C2 ZD2 = E2
(2.9)
It follows from Lemma 2.1 that matrix equation (2.8) is consistent if and only if
RA3 C3 = 0,
C3 LB3 = 0.
And the general solution to the matrix equations (2.8) can be expressed as
X1 = A†1 (E1 − C1 ZD1 ) − A†1W1 B1 + LA1 W2 , Y1 = RA1 (E1 − C1 ZD1 )B†1 + A1 A†1W1 + W3 RB1 , Z = A†3C3 B†3 + LA3 U + V RB3 ,
(2.10)
where W1 , W2 , W3 , U and V are arbitrary matrices over H with appropriate sizes. Substituting (2.10) into (2.9) gives
A2 X2 + Y2 B2 + A4UD2 + C2V B4 = C4 .
(2.11)
Hence, the system (1.4) is consistent if and only if the matrix equations (2.8) and (2.11) are consistent, respectively. By Lemma 2.1, we know that the matrix equation (2.11) is consistent if and only if
RM RA E = 0, ELB LN = 0, RA ELD = 0, RC ELB = 0. We know by Lemma 2.1 that the general solution of Eq. (2.11) can be expressed as (2.4)–(2.7). Let Bi vanish and Ci = I in Theorem 2.1, we can give some solvability conditions and general solution to the generalized mixed Sylvester matrix equations (1.2). Corollary 2.1 ([27]). Let Ai , Bi and Ci (i = 1, 2) be given. Set
A = RB1 B2 , B = RA2 A1 , C = B2 LA , D = RA2 (C2 − RA1 C1 B†1 B2 )LA . Then the generalized mixed Sylvester real quaternion matrix equations (1.2) is consistent if and only if
RA1 C1 LB1 = 0, RB D = 0, DLC = 0. In this case, the general solution to (1.2) can be expressed as
X1 = A†1C1 − U1 B1 + LA1 W1 , Z = RA1 C1 B†1 + A1U1 + V1 RB1 , X2 = A†2 (C2 − RA1 C1 B†1 B2 − A1U1 B2 ) − W4 A + LA2 W6 , where
U1 = B† DC † + LBW2 + W3 RC , V1 = RA2 (C2 − RA1 C1 B†1 B2 − A1U1 B2 )A† + A2W4 + W5 RA , and W1 , . . . , W6 are arbitrary matrices over H with appropriate sizes. Let B1 , A2 vanish and C1 = I, D2 = I in Theorem 2.1, we can present some solvability conditions and general solution to the generalized mixed Sylvester matrix equations (1.3). Corollary 2.2 ([14]). Let Ai , Bi and Ci (i = 1, 2) be given. Set
A = R(A2 A1 ) A2 , B = RB1 LB2 , C = R(A2 A1 ) (C2 − A2 RA1 C1 B†1 )LB2 . Then the generalized mixed Sylvester real quaternion matrix equations (1.3) is consistent if and only if
RA1 C1 LB1 = 0, RAC = 0, CLB = 0. In this case, the general solution to the mixed generalized Sylvester matrix equations (1.3) can be expressed as
X1 = A†1C1 − U1 B1 + LA1 W1 , Z = RA1 C1 B†1 + A1U1 + V1 RB1 , X2 = R(A2 A1 ) (C2 − A2 RA1 C1 B†1 − A2V1 RB1 )B†2 + A2 A1W4 + W5 RB2 , where
V1 = A†CB† + LAW2 + W3 RB , U1 = (A2 A1 )† (C2 − A2 RA1 C1 B†1 − A2V1 RB1 ) − W4 B2 + L(A2 A1 )W6 , and W1 , . . . , W6 are arbitrary matrices over H with appropriate sizes.
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X. Zhang / Applied Mathematics and Computation 273 (2016) 74–81
3. The solution to (1.5) with Z being η-Hermitian Based on Theorem 2.1, we can derive some solvability conditions and general solution to the real quaternion matrix equation (1.5) with Z being η-Hermitian. Lemma 3.1 ([16]). Let A ∈ Hm×n be given. Then
(Aη )† (A† A)η∗ (AA† )η∗ (LA )η∗ (RA )η∗
= (A† )η , (Aη∗ )† = (A† )η∗ .
= Aη∗ (A† )η∗ = (A† A)η = (A† )η Aη . = (A† )η∗ Aη∗ = (AA† )η = Aη (A† )η . = −η(LA )η = (LA )η = LAη = RAη∗ .
= −η(RA )η = (RA )η = RAη = LAη∗ .
Now we give the main Theorem of this section. Theorem 3.1. Let A1 , B1 , C1 , D1 and E1 be given. Set η∗
η∗
A2 = RA1 C1 , B2 = D1 LB1 , C2 = RA1 E1 LB1 , A3 = D1 LA2 , B3 = RB2 C1 , η∗
η∗
η∗
η∗
η∗
C3 = E1 − D1 A†2C2 B†2C1 , A = (LB1 )η∗ A3 , B = C1 (RA1 )η∗ , C = (LB1 )η∗ D1 , D = B3 (RA1 )η∗ , E = (LB1 )η∗C3 (RA1 )η∗ , M = RAC, N = DLB , S = CLM . Then the generalized Sylvester real quaternion matrix equation (1.5) has a solution (X, Y, Z), where Z is η-Hermitian, if and only if
RA2 C2 = 0, C2 LB2 = 0, RA ELD = 0. In this case, the solution to (1.5) can be expressed as η∗
X=
X1 + X2 , 2
η∗
Y =
Y1 + Y2 , 2
Z = Z η∗ =
η∗
Z1 + Z1 , 2
where
Z1 = A†2C2 B†2 + LA2 U + V RB2 , X1 = A†1 (E1 − C1 Z1 D1 ) − A†1W1 B1 + LA1 W2 , Y1 = RA1 (E1 − C1 Z1 D1 )B†1 + A1 A†1W1 + W3 RB1 , η∗
η∗
η∗
Y2 = (B†1 )η∗ (C3 − A3UC1 − D1 V B3 ) − (B†1 )η∗W4 A1 + (RB1 )η∗W5 , η∗
η∗
η∗
X2 = (LB1 )η∗ (C3 − A3UC1 − D1 V B3 )(A†1 )η∗ + B1 (B†1 )η∗W4 + W6 (LA1 )η∗ , U = A† EB† − A†CM† EB† − A† SC † EN† DB† − A† SW7 RN DB† + LAW8 + W9 RB , V = M† ED† + S† SC † EN† + LM LSW10 + LMW7 RN + W11 RD , and W1 , . . . , W11 are arbitrary matrices over H with appropriate sizes. Proof. We first show that the generalized Sylvester real quaternion matrix equation (1.5) has a solution (X, Y, Z = Z η∗ ), if and only if the system
A1 X1 + Y1 B1 + C1 Z1 D1 = E1 , η∗
η∗
η∗
η∗
(3.1)
η∗
B1 Y2 + X2 A1 + D1 Z1C1 = E1
η∗
has a solution. If the matrix equation (1.5) has a solution, say, (X0 , Y0 , Z0 = Z0 ), then
(X1 , Y1 , X2 , Y2 , Z1 ) = (X0 , Y0 , X0η∗ , Y0η∗ , Z0 ) is a solution to the system (3.1). Conversely, if the system (3.1) has a solution (X1 , Y1 , X2 , Y2 , Z1 ), check that
(X, Y, Z ) =
η∗
η∗
η∗
Y1 + Y2 Z1 + Z1 X1 + X2 , , 2 2 2
is a solution of (1.5). We can derive the solvability conditions and the expression of general solution by Theorem 2.1 and Lemma 3.1. Let A1 and B1 vanish in Theorem 3.1, we can present some solvability conditions and η-Hermitian solution to the real quaternion matrix equation
C1 ZD1 = E1 .
(3.2)
X. Zhang / Applied Mathematics and Computation 273 (2016) 74–81
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Corollary 3.1. Let C1 , D1 and E1 be given. Set η∗
η∗
η∗
η∗
η∗
A = D1 LC1 , D = RD1 C1 , E = E1 − D1 C1† E1 D†1C1 , η∗
η∗
M = RA D1 , N = B3 (RC1 )η∗ , S = D1 LM . Then the real quaternion matrix equation (3.2) has an η-Hermitian solution Z if and only if
RC1 E1 = 0, E1 LD1 = 0, RA ELD = 0. In this case, the η-Hermitian solution to (3.2) can be expressed as
Z = Z η∗ =
η∗
Z1 + Z1 , 2
where
Z1 = C1† E1 D†1 + LC1 U + V RD1 , η∗
U = A† E (C1† )η∗ − A† D1 M† E (C1† )η∗ − A† S(D†1 )η∗ EN† D(C1† )η∗ − A† SW1 RN D(C1† )η∗ + LAW2 + W3 (LC1 )η∗ , V = M† ED† + S† S(D†1 )η∗ EN† + LM LSW4 + LMW1 RN + W5 RD , and W1 , . . . , W5 are arbitrary matrices over H with appropriate sizes. 4. Solvability conditions and general solution to (1.6) Based on Theorem 2.1, we can obtain some solvability conditions and general solution to system of real quaternion matrix equations involving η-Hermicity (1.6). η∗
Theorem 4.1. Let Ai , Bi and Ci = Ci
(i = 1, 2) be given. Set
A3 = RA1 B1 , C3 = RA1 C1 (RA1 )η∗ , A4 = B2 LA3 , C4 = C2 − B2 A†3C3 (B2 A†3 )η∗ , A = RA2 A4 , B = (RA2 B2 )η∗ , E = RA2 C4 (RA2 )η∗ , M = RA Bη∗ , N = Aη∗ LB , S = Bη∗ LM . Then the system of real quaternion matrix equations (1.6) is consistent if and only if
RA3 C3 = 0, RM RA E = 0, RA E (RA )η∗ = 0. In this case, the general solution to (1.6) can be expressed as η∗
X=
X1 + Y1 , 2
η∗
Z=
X2 + Y2 , 2
Y = Y η∗ =
η∗
Z1 + Z1 , 2
where
Z1 = A†3C3 (A†3 )η∗ + LA3 U + V (LA3 )η∗ , η∗
X1 = A†1 (C1 − B1 Z1 B1
) − A†1W1 A1η∗ + LA1 W2 ,
η∗
Y1 = RA1 (C1 − B1 Z1 B1
)(A†1 )η∗ + A1 A†1W1 + W3 (LA1 )η∗ ,
η∗
η∗
η∗
η∗
X2 = A†2 (C4 − A4UB2 − B2VA4
) − A†2W4 A2η∗ + LA2 W5 ,
Y2 = RA2 (C4 − A4UB2 − B2VA4
)(A†2 )η∗ + A2 A†2W4 + W6 (LA2 )η∗ ,
U = A† EB† − A† Bη∗ M† EB† − A† S(B† )η∗ EN† Aη∗ B† − A† SW7 RN Aη∗ B† + LAW8 + W9 RB , V = M† E (A† )η∗ + S† S(B† )η∗ EN† + LM LSW10 + LMW7 RN + W11 (LA )η∗ , and W1 , . . . , W11 are arbitrary matrices over H with appropriate sizes. Proof. The proof is similar to that of Theorem 3.1. We first prove that the system of real quaternion matrix equations (1.6) has a solution if and only if the system
η∗
η∗
η∗
η∗
A1 X1 + Y1 A1 + B1 Z1 B1 = C1 ,
(4.1)
A2 X2 + Y2 A2 + B2 Z1 B2 = C2 has a solution. If the system (1.6) has a solution, say, (X0 , Z0 , Y0 ), then
(X1 , Y1 , X2 , Y2 , Z1 ) = (X0 , X0η∗ , Z0 , Z0η∗ , Y0 ) is a solution to the system (4.1). Conversely, if the system (4.1) has a solution (X1 , Y1 , X2 , Y2 , Z1 ), check that
(X, Z, Y ) =
η∗
η∗
η∗
X2 + Y2 Z1 + Z1 X1 + Y1 , , 2 2 2
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X. Zhang / Applied Mathematics and Computation 273 (2016) 74–81
is a solution of (1.6). We can derive the solvability conditions and the expression of general solution by Theorem 2.1 and Lemma 3.1. Let A1 vanish in Theorem 4.1, we can obtain some solvability conditions and η-Hermitian solution to the system of real quaternion matrix equations
η∗
B1Y B1 = C1 ,
(4.2)
η∗
B2Y B2 = C2 . η∗
Corollary 4.1. Let Bi and Ci = Ci
(i = 1, 2) be given. Set
A = B2 LB1 , C = C2 − B2 B†1C1 (B2 B†1 )η∗ , M = RA B2 , N = Aη∗ (RB2 )η∗ , S = B2 LM . Then the system of real quaternion matrix equations (4.2) has an η-Hermitian solution Y if and only if
RB1 C1 = 0, RM RNC = 0, RAC (RA )η∗ = 0. In this case, the general solution to (4.2) can be expressed as
Y = Y η∗ =
η∗
Z1 + Z1 , 2
where
Z1 = B†1C1 (B†1 )η∗ + LB1 U + V (LB1 )η∗ , U = A†C (B†2 )η∗ − A† B2 M†C (B†2 )η∗ − A† SB†2CN† Aη∗ (B†2 )η∗ − A† SW1 RN Aη∗ (B†2 )η∗ + LAW2 + W3 (LB2 )η∗ , V = M†C (A† )η∗ + S† SB†2CN† + LM LSW4 + LMW1 RN + W5 (LA )η∗ , and W1 , . . . , W5 are arbitrary matrices over H with appropriate sizes. 5. Conclusions We have derived some necessary and sufficient conditions for the existence and the expression of the general solution to system of generalized Sylvester real quaternion matrix equations (1.4). As special cases, the corresponding results on generalized mixed Sylvester matrix equations (1.2) and (1.3) are provided. Using the results on system (1.4), we have given some necessary and sufficient conditions for the solvability to the generalized Sylvester real quaternion matrix equation (1.5), where Z is required to be η-Hermitian. The expression of such a solution to (1.5) has also been given when its solvability conditions are met. We have presented some solvability conditions and general solution for the system of real quaternion matrix equations involving η-Hermicity (1.6). Some known results can be viewed as special cases of this paper. Acknowledgments This research was supported by the National Natural Science Foundation of China [11401125], Fund of Science and Technology Department of Guizhou Province LH [2014]7062, and the Doctor Fund of Guizhou Normal University. References [1] O. Axelsson, Z.-Z. Bai, S.-X. Qiu, A class of nested iteration schemes for linear systems with a coefficient matrix with a dominant positive definite symmetric part, Numer. Algorithms 35 (2004) 351–372. [2] Z.-Z. Bai, On Hermitian and skew-Hermitian splitting iteration methods for continuous Sylvester equations, J. Comput. Math. 29 (2011) 185–198. [3] J.K. Baksalary, R. Kala, The matrix equation AX − Y B = C, Linear Algebra Appl. 25 (1979) 41–43. [4] N.L. Bihan, J. Mars, Singular value decomposition of quaternion matrices: A new tool for vector-sensor signal processing, Signal Process. 84 (2004) 1177– 1199. [5] G.-R. Duan, Eigenstructure assignment and response analysis in descriptor linear systems with state feedback control, Int. J. Control 69 (1998) 663–694. [6] Y.-B. Deng, Z.-Z. Bai, Y.-H. Gao, Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations, Numer. Linear Algebra Appl. 13 (2006) 801–823. [7] L.R. Fletcher, J. Kautsky, N.K. Nichols, Eigenstructure assignment in descriptor systems, IEEE Trans. Automat. Control 31 (1986) 1138–1141. [8] X. Li, Y.-J. Wu, A.-L. Yang, J.-Y. Yuan, A generalized HSS iteration method for continuous Sylvester equations, J. Appl. Math. (2014). Art. ID 578102 [9] A.-P. Liao, Z.-Z. Bai, Least-squares solution of AXB = D over symmetric positive semidefinite matrices X, J. Comput. Math. 21 (2003) 175–182. [10] A.-P. Liao, Z.-Z. Bai, Least squares symmetric and skew-symmetric solutions of the matrix equation AXAT + BY BT = C with the least norm, Math. Numer. Sinica 27 (2005) 81–95. (In Chinese) [11] A.-P. Liao, Z.-Z. Bai, Y. Lei, Best approximate solution of matrix equation AXB + CY D = E, SIAM J. Matrix Anal. Appl. 27 (2005) 675–688. [12] Q. Niu, X. Wang, L.-Z. Lu, A relaxed gradient based algorithm for solving Sylvester equations, Asian J. Control 13 (2011) 461–464. [13] Z.-H. He, Q.-W. Wang, The η-bihermitian solution to a system of real quaternion matrix equations, Linear Multilinear Algebra 62 (2014) 1509–1528. [14] Z.-H. He, Q.-W. Wang, A pair of mixed generalized Sylvester matrix equations, J. Shanghai Univ. Nat. Sci. 20 (2014) 138–156. [15] Z.-H. He, Q.-W. Wang, The general solutions to some systems of matrix equations, Linear Multilinear Algebra 63 (2015) 2017–2032. [16] Z.-H. He, Q.-W. Wang, A real quaternion matrix equation with applications, Linear Multilinear Algebra 61 (2013) 725–740. [17] R.A. Horn, F.-Z. Zhang, A generalization of the complex Autonne–Takagi factorization to quaternion matrices, Linear Multilinear Algebra 60 (2012) 1239– 1244.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A system of generalized Sylvester quaternion matrix equations and its applications Xiang Zhang∗ Department of Mathematics, Guizhou Normal University, Guiyang 550001, PR China
a r t i c l e
i n f o
MSC: 15A03 15A09 15A24 11R52 15B57 Keywords: System of matrix equation Quaternion General solution η-Hermitian Moore–Penrose inverse
a b s t r a c t Let Hm×n be the set of all m × n matrices over the real quaternion algebra. We call that A ∈ Hn×n is η-Hermitian if A = −ηA∗ η, η ∈ {i, j, k}, where i, j, k are the quaternion units. Denote Aη∗ = −ηA∗ η. In this paper, we derive some necessary and sufficient conditions for the solvability to the system of generalized Sylvester real quaternion matrix equations Ai Xi + Yi Bi + Ci ZDi = Ei , (i = 1, 2), and give an expression of the general solution to the above mentioned system. As applications, we give some solvability conditions and general solution for the generalized Sylvester real quaternion matrix equation A1 X + Y B1 + C1 ZD1 = E1 , where Z is required to be η-Hermitian. We also present some solvability conditions and general solution for the sysη∗ tem of real quaternion matrix equations involving η-Hermicity Ai Xi + (Ai Xi )η∗ + BiY Bi = Ci , (i = 1, 2), where Y is required to be η-Hermitian. Our results include some well-known results as special cases. © 2015 Elsevier Inc. All rights reserved.
1. Introduction Throughout the paper, the set of all m × n matrices over the quaternion number field H
H = {a0 + a1 i + a2 j + a3 k | i2 = j2 = k2 = i jk = −1, a0 , a1 , a2 , a3 ∈ R} is denoted by Hm×n . For a matrix A, A∗ stands for the conjugate transpose of A. I denotes the identity matrix with appropriate sizes. The Moore–Penrose inverse A† of A, is defined to be the unique matrix A† , such that
(i) AA† A = A, (ii) A† AA† = A† , (iii) (AA† )∗ = AA† , (iv) (A† A)∗ = A† A. Furthermore, LA and RA stand for the two projectors LA = I − A† A and RA = I − AA† induced by A, respectively. It is known that LA = L∗A and RA = R∗A . Quaternions were introduced by Irish mathematician Sir William Rowan Hamilton in 1843. Quaternions are associative and noncommutative. Nowadays quaternion matrices have been widely and heavily used in computer science, quantum physics, signal and color image processing, and so on (e.g. [4,21,22,24,25,38]). Many problems can be reduced to solving systems of quaternion matrix equations. Many problems in systems and control theory require the solution of Sylvester matrix equation. The Sylvester matrix equation has a huge amount of practical applications in feedback control ([5,18,32,36,37]), robust control [20], pole/eigenstructure assignment design [7], neural network [34], and so on. ∗
Corresponding author. Tel.: +14785520175. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.amc.2015.09.074 0096-3003/© 2015 Elsevier Inc. All rights reserved.
X. Zhang / Applied Mathematics and Computation 273 (2016) 74–81
75
Roth [23] gave a necessary and sufficient condition for the solvability to the generalized Sylvester matrix equation
A1 X + Y B1 = C1 .
(1.1)
Bakasalary and Kala [3] derived the general solution to the generalized Sylvester matrix equation (1.1) by using generalized inverses. Bai [2] considerd iterative solutions of the continuous Sylvester equations of the form AX + XB = C. Li et al. [8] established a generalized HSS (GHSS) iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and positive definite/semidefinite matrices. Lin and Wei [31] investigated the condition numbers of the generalized Sylvester equation of the form
AX − Y B = C, DX − Y E = F.
In recent years, the study on mixed generalized Sylvester matrix equations are active ([1,6,9–12,29,30,39]). Lee and Vu [19] considered the generalized mixed Sylvester matrix equations
A1 X1 + ZB1 = C1 ,
(1.2)
A2 X2 + ZB2 = C2 ,
where Ai , Bi and Ci are given complex matrices. They gave a necessary and sufficient condition for the solvability to the generalized mixed Sylvester matrix equations (1.2). Wang and He [27] presented necessary and sufficient solvability conditions for the system (1.2), and gave general solution when it is solvable. Recently, He and Wang [14] considered the other form of the generalized mixed Sylvester matrix equations, i.e.,
A1 X1 + ZB1 = C1 ,
(1.3)
A2 Z + X2 B2 = C2 ,
where Ai , Bi and Ci are given complex matrices. He and Wang presented some necessary and sufficient solvability conditions for the system (1.3) and derived an expression of the general solution to (1.3). Motivated by the work mentioned above and the recent increasing interests in quaternion matrices and generalized mixed Sylvester matrix equations, we consider the following more general system of generalized Sylvester real quaternion matrix equations
A1 X1 + Y1 B1 + C1 ZD1 = E1 ,
(1.4)
A2 X2 + Y2 B2 + C2 ZD2 = E2 ,
where Ai , Bi , Ci , Di and Ei (i = 1, 2) are given real quaternion matrices. The generalized Sylvester matrix equation (1.1), mixed Sylvester matrix equations (1.2) and (1.3) are special cases of the system (1.4). One contribution of this paper is to give some solvability conditions and general solution to the system of generalized Sylvester real quaternion matrix equations (1.4). Recall that a quaternion matrix A is called an η-Hermitian matrix if A = Aη∗ = −ηA∗ η, and η ∈ {i, j, k} [26]. The η-Hermitian matrices have some applications widely used in linear modelling, and convergence analysis in statistical signal processing [25]. Horn and Zhang [17] presented an analogous special singular value decomposition for η-Hermitian matrices. Yuan and Wang [33] gave the expressions of the least squares η-Hermitian solution of the real quaternion matrix equation AXB + CXD = E. He and Wang [16] derived some solvability conditions and general solution to the real quaternion matrix equation involving η-Hermicity, i.e., η∗
η∗
A1 X + (A1 X )η∗ + B1Y B1 + C1 ZC1 = D1 , where Y and Z are required to be η-Hermitian. They investigated some properties of the quaternion matrix Aη∗ . Zhang and Wang [35] provided some solvability conditions and the general solution to a system of quaternion matrix equations involving η-Hermicity:
A1 X = C1 , A2Y = C2 , ZB2 = D2 , η∗
η∗
A3 X + (A3 X )η∗ + B3Y B3 + C3 ZC3 = D3 , where Y = Y η∗ , Z = Z η∗ . Recently, He and Wang [13] considered the η-Hermitian solution to a system of real quaternion matrix equations
⎧ ⎨A1 X = C1 , XB1 = D1 , A2Y = C2 , Y B2 = D2 ,
⎩
η∗
η∗
C3 XC3 + D3Y D3 = A3 .
As applications of the system (1.4), we can consider the following two real quaternion matrix equations involving η-Hermicity
A1 X + Y B1 + C1 ZD1 = E1 , and
Z = Z η∗
(1.5)
η∗
A1 X + (A1 X )η∗ + B1Y B1 = C1 , η∗ A2 Z + (A2 Z )η∗ + B2Y B2 = C2 ,
(1.6)
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where Y = Y η∗ . The other goal of this paper is to give some solvability conditions and general solution to the real quaternion matrix equations (1.5) and (1.6), respectively. The remainder of the paper is organized as follows. In Section 2, we establish some necessary and sufficient conditions for the existence of a solution to the system of generalized Sylvester real quaternion matrix equations (1.4) and derive the explicit expression of the general solution to system (1.4). In Sections 3 and 4, as applications, we give some solvability conditions and the general solutions to real quaternion matrix equations involving η-Hermicity, i.e., the generalized Sylvester real quaternion matrix equation (1.5) and the system of real quaternion matrix equations (1.6), respectively. 2. Some solvability conditions and the general solution to the system (1.4) In this section, using the inverses of the coefficient matrices, we give some solvability conditions for the system of generalized Sylvester real quaternion matrix equations (1.4) to possess a solution and to present an expression of this general solution when it is solvable. We begin with the following lemma. Lemma 2.1 ([15,28]). Let A1 ∈ Hm×n1 , B1 ∈ H p1 ×q , C3 ∈ Hm×n2 , D3 ∈ H p2 ×q , C4 ∈ Hm×n3 , D4 ∈ H p3 ×q , and E1 ∈ Hm×q be given. Set
A = RA1 C3 , B = D3 LB1 , C = RA1 C4 , D = D4 LB1 , E = RA1 E1 LB1 , M = RAC, N = DLB , S = CLM . Then the equation
A1 X1 + X2 B1 + C3 X3 D3 + C4 X4 D4 = E1
(2.1)
is consistent if and only if
RM RA E = 0,
ELB LN = 0,
RA ELD = 0,
RC ELB = 0.
(2.2)
In this case, the general solution of (2.1) can be expressed as
X1 = A†1 (E1 − C3 X3 D3 − C4 X4 D4 ) − A†1W2 B1 + LA1 W1 , X2 = RA1 (E1 − C3 X3 D3 − C4 X4 D4 )B†1 + A1 A†1W2 + W3 RB1 , X3 = A† EB† − A†CM† EB† − A† SC † EN† DB† − A† SV4 RN DB† + LAV3 + V4 RB , X4 = M† ED† + S† SC † EN† + LM LSU1 + LMV4 RN + V5 RD , where V1 , V2 , V3 , V4 , V5 , W1 , W2 , W3 are arbitrary matrices over H with appropriate sizes. Now we give the fundamental theorem of this paper. Theorem 2.1. Let Ai , Bi , Ci , Di and Ei (i = 1, 2) be given. Set
A3 = RA1 C1 , B3 = D1 LB1 , C3 = RA1 E1 LB1 , A4 = C2 LA3 , B4 = RB3 D2 , C4 = E2 − C2 A†3C3 B†3 D2 , A = RA2 A4 , B = D2 LB2 , C = RA2 C2 , D = B4 LB2 , E = RA2 C4 LB2 , M = RAC, N = DLB , S = CLM . Then the system of generalized Sylvester real quaternion matrix equations (1.4) is consistent if and only if
RA3 C3 = 0, C3 LB3 = 0, RM RA E = 0, ELB LN = 0, RA ELD = 0, RC ELB = 0.
(2.3)
In this case, the general solution of (1.4) can be expressed as
Z = A†3C3 B†3 + LA3 U + V RB3 , X1 = A†1 (E1 − C1 ZD1 ) − A†1W1 B1 + LA1 W2 , Y1 = RA1 (E1 − C1 ZD1 )B†1 + A1 A†1W1 + W3 RB1 , X2 = A†2 (C4 − A4UD2 − C2V B4 ) − A†2W4 B2 + LA2 W5 ,
(2.4)
Y2 = RA2 (C4 −
(2.5)
)
A4UD2 − C2V B4 B†2
+
A2 A†2W4
+ W6 RB2 ,
where
U = A† EB† − A†CM† EB† − A† SC † EN† DB† − A† SW7 RN DB† + LAW8 + W9 RB ,
(2.6)
V = M† ED† + S† SC † EN† + LM LSW10 + LMW7 RN + W11 RD ,
(2.7)
and W1 , . . . , W11 are arbitrary matrices over H with appropriate sizes.
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77
Proof. We separate the equations in system (1.4) into two groups
A1 X1 + Y1 B1 + C1 ZD1 = E1 ,
(2.8)
A2 X2 + Y2 B2 + C2 ZD2 = E2
(2.9)
It follows from Lemma 2.1 that matrix equation (2.8) is consistent if and only if
RA3 C3 = 0,
C3 LB3 = 0.
And the general solution to the matrix equations (2.8) can be expressed as
X1 = A†1 (E1 − C1 ZD1 ) − A†1W1 B1 + LA1 W2 , Y1 = RA1 (E1 − C1 ZD1 )B†1 + A1 A†1W1 + W3 RB1 , Z = A†3C3 B†3 + LA3 U + V RB3 ,
(2.10)
where W1 , W2 , W3 , U and V are arbitrary matrices over H with appropriate sizes. Substituting (2.10) into (2.9) gives
A2 X2 + Y2 B2 + A4UD2 + C2V B4 = C4 .
(2.11)
Hence, the system (1.4) is consistent if and only if the matrix equations (2.8) and (2.11) are consistent, respectively. By Lemma 2.1, we know that the matrix equation (2.11) is consistent if and only if
RM RA E = 0, ELB LN = 0, RA ELD = 0, RC ELB = 0. We know by Lemma 2.1 that the general solution of Eq. (2.11) can be expressed as (2.4)–(2.7). Let Bi vanish and Ci = I in Theorem 2.1, we can give some solvability conditions and general solution to the generalized mixed Sylvester matrix equations (1.2). Corollary 2.1 ([27]). Let Ai , Bi and Ci (i = 1, 2) be given. Set
A = RB1 B2 , B = RA2 A1 , C = B2 LA , D = RA2 (C2 − RA1 C1 B†1 B2 )LA . Then the generalized mixed Sylvester real quaternion matrix equations (1.2) is consistent if and only if
RA1 C1 LB1 = 0, RB D = 0, DLC = 0. In this case, the general solution to (1.2) can be expressed as
X1 = A†1C1 − U1 B1 + LA1 W1 , Z = RA1 C1 B†1 + A1U1 + V1 RB1 , X2 = A†2 (C2 − RA1 C1 B†1 B2 − A1U1 B2 ) − W4 A + LA2 W6 , where
U1 = B† DC † + LBW2 + W3 RC , V1 = RA2 (C2 − RA1 C1 B†1 B2 − A1U1 B2 )A† + A2W4 + W5 RA , and W1 , . . . , W6 are arbitrary matrices over H with appropriate sizes. Let B1 , A2 vanish and C1 = I, D2 = I in Theorem 2.1, we can present some solvability conditions and general solution to the generalized mixed Sylvester matrix equations (1.3). Corollary 2.2 ([14]). Let Ai , Bi and Ci (i = 1, 2) be given. Set
A = R(A2 A1 ) A2 , B = RB1 LB2 , C = R(A2 A1 ) (C2 − A2 RA1 C1 B†1 )LB2 . Then the generalized mixed Sylvester real quaternion matrix equations (1.3) is consistent if and only if
RA1 C1 LB1 = 0, RAC = 0, CLB = 0. In this case, the general solution to the mixed generalized Sylvester matrix equations (1.3) can be expressed as
X1 = A†1C1 − U1 B1 + LA1 W1 , Z = RA1 C1 B†1 + A1U1 + V1 RB1 , X2 = R(A2 A1 ) (C2 − A2 RA1 C1 B†1 − A2V1 RB1 )B†2 + A2 A1W4 + W5 RB2 , where
V1 = A†CB† + LAW2 + W3 RB , U1 = (A2 A1 )† (C2 − A2 RA1 C1 B†1 − A2V1 RB1 ) − W4 B2 + L(A2 A1 )W6 , and W1 , . . . , W6 are arbitrary matrices over H with appropriate sizes.
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3. The solution to (1.5) with Z being η-Hermitian Based on Theorem 2.1, we can derive some solvability conditions and general solution to the real quaternion matrix equation (1.5) with Z being η-Hermitian. Lemma 3.1 ([16]). Let A ∈ Hm×n be given. Then
(Aη )† (A† A)η∗ (AA† )η∗ (LA )η∗ (RA )η∗
= (A† )η , (Aη∗ )† = (A† )η∗ .
= Aη∗ (A† )η∗ = (A† A)η = (A† )η Aη . = (A† )η∗ Aη∗ = (AA† )η = Aη (A† )η . = −η(LA )η = (LA )η = LAη = RAη∗ .
= −η(RA )η = (RA )η = RAη = LAη∗ .
Now we give the main Theorem of this section. Theorem 3.1. Let A1 , B1 , C1 , D1 and E1 be given. Set η∗
η∗
A2 = RA1 C1 , B2 = D1 LB1 , C2 = RA1 E1 LB1 , A3 = D1 LA2 , B3 = RB2 C1 , η∗
η∗
η∗
η∗
η∗
C3 = E1 − D1 A†2C2 B†2C1 , A = (LB1 )η∗ A3 , B = C1 (RA1 )η∗ , C = (LB1 )η∗ D1 , D = B3 (RA1 )η∗ , E = (LB1 )η∗C3 (RA1 )η∗ , M = RAC, N = DLB , S = CLM . Then the generalized Sylvester real quaternion matrix equation (1.5) has a solution (X, Y, Z), where Z is η-Hermitian, if and only if
RA2 C2 = 0, C2 LB2 = 0, RA ELD = 0. In this case, the solution to (1.5) can be expressed as η∗
X=
X1 + X2 , 2
η∗
Y =
Y1 + Y2 , 2
Z = Z η∗ =
η∗
Z1 + Z1 , 2
where
Z1 = A†2C2 B†2 + LA2 U + V RB2 , X1 = A†1 (E1 − C1 Z1 D1 ) − A†1W1 B1 + LA1 W2 , Y1 = RA1 (E1 − C1 Z1 D1 )B†1 + A1 A†1W1 + W3 RB1 , η∗
η∗
η∗
Y2 = (B†1 )η∗ (C3 − A3UC1 − D1 V B3 ) − (B†1 )η∗W4 A1 + (RB1 )η∗W5 , η∗
η∗
η∗
X2 = (LB1 )η∗ (C3 − A3UC1 − D1 V B3 )(A†1 )η∗ + B1 (B†1 )η∗W4 + W6 (LA1 )η∗ , U = A† EB† − A†CM† EB† − A† SC † EN† DB† − A† SW7 RN DB† + LAW8 + W9 RB , V = M† ED† + S† SC † EN† + LM LSW10 + LMW7 RN + W11 RD , and W1 , . . . , W11 are arbitrary matrices over H with appropriate sizes. Proof. We first show that the generalized Sylvester real quaternion matrix equation (1.5) has a solution (X, Y, Z = Z η∗ ), if and only if the system
A1 X1 + Y1 B1 + C1 Z1 D1 = E1 , η∗
η∗
η∗
η∗
(3.1)
η∗
B1 Y2 + X2 A1 + D1 Z1C1 = E1
η∗
has a solution. If the matrix equation (1.5) has a solution, say, (X0 , Y0 , Z0 = Z0 ), then
(X1 , Y1 , X2 , Y2 , Z1 ) = (X0 , Y0 , X0η∗ , Y0η∗ , Z0 ) is a solution to the system (3.1). Conversely, if the system (3.1) has a solution (X1 , Y1 , X2 , Y2 , Z1 ), check that
(X, Y, Z ) =
η∗
η∗
η∗
Y1 + Y2 Z1 + Z1 X1 + X2 , , 2 2 2
is a solution of (1.5). We can derive the solvability conditions and the expression of general solution by Theorem 2.1 and Lemma 3.1. Let A1 and B1 vanish in Theorem 3.1, we can present some solvability conditions and η-Hermitian solution to the real quaternion matrix equation
C1 ZD1 = E1 .
(3.2)
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Corollary 3.1. Let C1 , D1 and E1 be given. Set η∗
η∗
η∗
η∗
η∗
A = D1 LC1 , D = RD1 C1 , E = E1 − D1 C1† E1 D†1C1 , η∗
η∗
M = RA D1 , N = B3 (RC1 )η∗ , S = D1 LM . Then the real quaternion matrix equation (3.2) has an η-Hermitian solution Z if and only if
RC1 E1 = 0, E1 LD1 = 0, RA ELD = 0. In this case, the η-Hermitian solution to (3.2) can be expressed as
Z = Z η∗ =
η∗
Z1 + Z1 , 2
where
Z1 = C1† E1 D†1 + LC1 U + V RD1 , η∗
U = A† E (C1† )η∗ − A† D1 M† E (C1† )η∗ − A† S(D†1 )η∗ EN† D(C1† )η∗ − A† SW1 RN D(C1† )η∗ + LAW2 + W3 (LC1 )η∗ , V = M† ED† + S† S(D†1 )η∗ EN† + LM LSW4 + LMW1 RN + W5 RD , and W1 , . . . , W5 are arbitrary matrices over H with appropriate sizes. 4. Solvability conditions and general solution to (1.6) Based on Theorem 2.1, we can obtain some solvability conditions and general solution to system of real quaternion matrix equations involving η-Hermicity (1.6). η∗
Theorem 4.1. Let Ai , Bi and Ci = Ci
(i = 1, 2) be given. Set
A3 = RA1 B1 , C3 = RA1 C1 (RA1 )η∗ , A4 = B2 LA3 , C4 = C2 − B2 A†3C3 (B2 A†3 )η∗ , A = RA2 A4 , B = (RA2 B2 )η∗ , E = RA2 C4 (RA2 )η∗ , M = RA Bη∗ , N = Aη∗ LB , S = Bη∗ LM . Then the system of real quaternion matrix equations (1.6) is consistent if and only if
RA3 C3 = 0, RM RA E = 0, RA E (RA )η∗ = 0. In this case, the general solution to (1.6) can be expressed as η∗
X=
X1 + Y1 , 2
η∗
Z=
X2 + Y2 , 2
Y = Y η∗ =
η∗
Z1 + Z1 , 2
where
Z1 = A†3C3 (A†3 )η∗ + LA3 U + V (LA3 )η∗ , η∗
X1 = A†1 (C1 − B1 Z1 B1
) − A†1W1 A1η∗ + LA1 W2 ,
η∗
Y1 = RA1 (C1 − B1 Z1 B1
)(A†1 )η∗ + A1 A†1W1 + W3 (LA1 )η∗ ,
η∗
η∗
η∗
η∗
X2 = A†2 (C4 − A4UB2 − B2VA4
) − A†2W4 A2η∗ + LA2 W5 ,
Y2 = RA2 (C4 − A4UB2 − B2VA4
)(A†2 )η∗ + A2 A†2W4 + W6 (LA2 )η∗ ,
U = A† EB† − A† Bη∗ M† EB† − A† S(B† )η∗ EN† Aη∗ B† − A† SW7 RN Aη∗ B† + LAW8 + W9 RB , V = M† E (A† )η∗ + S† S(B† )η∗ EN† + LM LSW10 + LMW7 RN + W11 (LA )η∗ , and W1 , . . . , W11 are arbitrary matrices over H with appropriate sizes. Proof. The proof is similar to that of Theorem 3.1. We first prove that the system of real quaternion matrix equations (1.6) has a solution if and only if the system
η∗
η∗
η∗
η∗
A1 X1 + Y1 A1 + B1 Z1 B1 = C1 ,
(4.1)
A2 X2 + Y2 A2 + B2 Z1 B2 = C2 has a solution. If the system (1.6) has a solution, say, (X0 , Z0 , Y0 ), then
(X1 , Y1 , X2 , Y2 , Z1 ) = (X0 , X0η∗ , Z0 , Z0η∗ , Y0 ) is a solution to the system (4.1). Conversely, if the system (4.1) has a solution (X1 , Y1 , X2 , Y2 , Z1 ), check that
(X, Z, Y ) =
η∗
η∗
η∗
X2 + Y2 Z1 + Z1 X1 + Y1 , , 2 2 2
80
X. Zhang / Applied Mathematics and Computation 273 (2016) 74–81
is a solution of (1.6). We can derive the solvability conditions and the expression of general solution by Theorem 2.1 and Lemma 3.1. Let A1 vanish in Theorem 4.1, we can obtain some solvability conditions and η-Hermitian solution to the system of real quaternion matrix equations
η∗
B1Y B1 = C1 ,
(4.2)
η∗
B2Y B2 = C2 . η∗
Corollary 4.1. Let Bi and Ci = Ci
(i = 1, 2) be given. Set
A = B2 LB1 , C = C2 − B2 B†1C1 (B2 B†1 )η∗ , M = RA B2 , N = Aη∗ (RB2 )η∗ , S = B2 LM . Then the system of real quaternion matrix equations (4.2) has an η-Hermitian solution Y if and only if
RB1 C1 = 0, RM RNC = 0, RAC (RA )η∗ = 0. In this case, the general solution to (4.2) can be expressed as
Y = Y η∗ =
η∗
Z1 + Z1 , 2
where
Z1 = B†1C1 (B†1 )η∗ + LB1 U + V (LB1 )η∗ , U = A†C (B†2 )η∗ − A† B2 M†C (B†2 )η∗ − A† SB†2CN† Aη∗ (B†2 )η∗ − A† SW1 RN Aη∗ (B†2 )η∗ + LAW2 + W3 (LB2 )η∗ , V = M†C (A† )η∗ + S† SB†2CN† + LM LSW4 + LMW1 RN + W5 (LA )η∗ , and W1 , . . . , W5 are arbitrary matrices over H with appropriate sizes. 5. Conclusions We have derived some necessary and sufficient conditions for the existence and the expression of the general solution to system of generalized Sylvester real quaternion matrix equations (1.4). As special cases, the corresponding results on generalized mixed Sylvester matrix equations (1.2) and (1.3) are provided. Using the results on system (1.4), we have given some necessary and sufficient conditions for the solvability to the generalized Sylvester real quaternion matrix equation (1.5), where Z is required to be η-Hermitian. The expression of such a solution to (1.5) has also been given when its solvability conditions are met. We have presented some solvability conditions and general solution for the system of real quaternion matrix equations involving η-Hermicity (1.6). Some known results can be viewed as special cases of this paper. Acknowledgments This research was supported by the National Natural Science Foundation of China [11401125], Fund of Science and Technology Department of Guizhou Province LH [2014]7062, and the Doctor Fund of Guizhou Normal University. References [1] O. Axelsson, Z.-Z. Bai, S.-X. Qiu, A class of nested iteration schemes for linear systems with a coefficient matrix with a dominant positive definite symmetric part, Numer. Algorithms 35 (2004) 351–372. [2] Z.-Z. Bai, On Hermitian and skew-Hermitian splitting iteration methods for continuous Sylvester equations, J. Comput. Math. 29 (2011) 185–198. [3] J.K. Baksalary, R. Kala, The matrix equation AX − Y B = C, Linear Algebra Appl. 25 (1979) 41–43. [4] N.L. Bihan, J. Mars, Singular value decomposition of quaternion matrices: A new tool for vector-sensor signal processing, Signal Process. 84 (2004) 1177– 1199. [5] G.-R. Duan, Eigenstructure assignment and response analysis in descriptor linear systems with state feedback control, Int. J. Control 69 (1998) 663–694. [6] Y.-B. Deng, Z.-Z. Bai, Y.-H. Gao, Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations, Numer. Linear Algebra Appl. 13 (2006) 801–823. [7] L.R. Fletcher, J. Kautsky, N.K. Nichols, Eigenstructure assignment in descriptor systems, IEEE Trans. Automat. Control 31 (1986) 1138–1141. [8] X. Li, Y.-J. Wu, A.-L. Yang, J.-Y. Yuan, A generalized HSS iteration method for continuous Sylvester equations, J. Appl. Math. (2014). Art. ID 578102 [9] A.-P. Liao, Z.-Z. Bai, Least-squares solution of AXB = D over symmetric positive semidefinite matrices X, J. Comput. Math. 21 (2003) 175–182. [10] A.-P. Liao, Z.-Z. Bai, Least squares symmetric and skew-symmetric solutions of the matrix equation AXAT + BY BT = C with the least norm, Math. Numer. Sinica 27 (2005) 81–95. (In Chinese) [11] A.-P. Liao, Z.-Z. Bai, Y. Lei, Best approximate solution of matrix equation AXB + CY D = E, SIAM J. Matrix Anal. Appl. 27 (2005) 675–688. [12] Q. Niu, X. Wang, L.-Z. Lu, A relaxed gradient based algorithm for solving Sylvester equations, Asian J. Control 13 (2011) 461–464. [13] Z.-H. He, Q.-W. Wang, The η-bihermitian solution to a system of real quaternion matrix equations, Linear Multilinear Algebra 62 (2014) 1509–1528. [14] Z.-H. He, Q.-W. Wang, A pair of mixed generalized Sylvester matrix equations, J. Shanghai Univ. Nat. Sci. 20 (2014) 138–156. [15] Z.-H. He, Q.-W. Wang, The general solutions to some systems of matrix equations, Linear Multilinear Algebra 63 (2015) 2017–2032. [16] Z.-H. He, Q.-W. Wang, A real quaternion matrix equation with applications, Linear Multilinear Algebra 61 (2013) 725–740. [17] R.A. Horn, F.-Z. Zhang, A generalization of the complex Autonne–Takagi factorization to quaternion matrices, Linear Multilinear Algebra 60 (2012) 1239– 1244.
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