Solutions of a system of mixed generalized Sylvester matrix equations

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1 Dec Communication on Applied Mathematics and Computation Vol.28 No.4 DOI /j.issn Solutions of a system of mixed generalized Sylvester matrix equations XU Huan 1, ZHANG Yang 2 (1. College of Sciences, Shanghai University, Shanghai , China; 2. Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada) Abstract In this paper, we derive some necessary and sufficient conditions for the existence of a solution to the system of mixed generalized Sylvester matrix equations A1 X = B 1, Y A 2 = B 2, A 3 Z = B 3, C 1 X Y D 1 = E 1, C 2 Z Y D 2 = E 2, where A i, B i, C j, D j, and E j (i = 1, 2, 3, j = 1, 2) are given complex matrices, and X, Y, and Z are variable matrices. We give an expression of the general solution to the above system when it is solvable. Moreover, we investigate the admissible ranks of the general solution to the system. This paper extends the known results in Wang and He s recent paper (Wang Q W, He Z H. Solvability conditions and general solution for mixed Sylvester equations. Automatica, 2013, 49: ). Key words Sylvester matrix equation; Moore-Penrose inverse; general solution 2010 Mathematics Subject Classification 15A24; 15A09; 15A03 Chinese Library Classification O ÇÆ Sylvester Ê Â ÁÎÉ 1, 2 (1. º º ; 2. ³µ ¼ µ MB R3T 2N2, ) Æ ± Sylvester ¹ ½ ű Sylvester ¾ ű Sylvester ± ² ÄÈÀ Sylvester Moore-Penrose ¾ 2010 Í ÃËÅ 15A24; 15A09; 15A03 Received ; Revised Corresponding author ZHANG Yang, research interests are computer algebra, analysis of matrix, and computation of polynomial. Yang.Zhang@umanitoba.ca

2 No. 4 XU Huan, et al.: Solutions of mixed generalized Sylvester matrix equations 403 ÏÃËÅ O Ì A Å (2014) Introduction Many problems in conventional linear control systems theory, such as neural network 1, robust control 2, feedback control 3-4, and pole/eigenstructure assignment design 5-6, are closely related with the generalized Sylvester matrix equation. Thus, many excellent work in this area have been done, for example, 2, 4, Recently, more and more people are getting interested in the mixed generalized Sylvester matrix equations A1 X Y B 1 = C 1, (1) A 2 Z Y B 2 = C 2, and A1 X Y B 1 = C 1, A 2 Y ZB 2 = C 2, (2) where A i, B i, and C i (i = 1, 2) are given complex matrices, X, Y, and Z are variable matrices. Lee and Vu 17 derived a necessary and sufficient solvability condition for the system (1). Liu 16 gave a solvability condition to (1). Very recently, Wang and He 23 considered a new necessary and sufficient solvability condition and the general solution to the mixed generalized Sylvester matrix equations (1). He and Wang 26 considered some necessary and sufficient solvability conditions and general solution to the system of mixed generalized Sylvester matrix equations (2). To our knowledge, there has been little information on the system of mixed generalized Sylvester matrix equations A1 X = B 1, Y A 2 = B 2, A 3 Z = B 3, C 1 X Y D 1 = E 1, C 2 Z Y D 2 = E 2. (3) Motivated by the work mentioned above, as well as the wide application of Sylvester matrix equations, we consider the general solution to the system of mixed generalized Sylvester matrix equations (3). The paper is organized as follows. In Section 1, we review some known results and lemmas. In Section 2, some solvability conditions and the general solution to the system (3) are given. In Section 3, we derive the admissible ranks of the general solution to the system (3). Throughout this paper, we denote the complex number field by C. The notations C m n and C m m h stand for the sets of all m n complex matrices and all m m complex Hermitian matrices, respectively. The identity matrix with an appropriate size is denoted by I. For

3 404 Communication on Applied Mathematics and Computation Vol. 28 a complex matrix A, the symbols A and r(a) stand for the conjugate transpose and the rank of A, respectively. The Moore-Penrose inverse of A C m n, denoted by A, is defined to be the unique solution X to the following four matrix equations: AXA = A, XAX = X, (AX) = AX, (XA) = XA. Furthermore, L A and R A stand for the two projectors L A = I A A and R A = I AA induced by A, respectively. It is known that L A = L A and R A = RA. 1 Preliminaries In this section, we give some preliminary results which will be used in the rest part of this paper. Lemma 1 19 Let A, B, and C be given with appropriate sizes. Then, the following statements are equivalent. (i) Equation AXB = C (4) is consistent; (ii) R A C = 0 and CL B = 0; (iii) C r A C = r(a), r = r(b). B In this case, the general solution to the equation (4) can be expressed as X = A CB + L A V 1 + V 2 R B, where V 1 and V 2 are arbitrary matrices over C with appropriate sizes. Lemma 2 23 Let A i, B i, and C i (i = 1, 2) be given. Set D 1 = R B1 B 2, A = R A2, B = B 2 L D1, C = R A2 (R A1 C 1 B 1 B 2 C 2 )L D1. Then, the following statements are equivalent. (i) The mixed Sylvester matrix equations (1) are consistent; (ii) R A1 C 1 L B1 = 0, R A C = 0, CL B = 0. In this case, the general solution to the mixed Sylvester matrix equations (1) can be expressed as X = A 1 C 1 + U 1 B 1 + L A1 W 1, Y = R A1 C 1 B 1 + U 1 + V 1 R B1, Z = A 2 (C 2 R A1 C 1 B 1 B 2 + U 1 B 2 ) + W 4 D 1 + L A2 W 6,

4 No. 4 XU Huan, et al.: Solutions of mixed generalized Sylvester matrix equations 405 where U 1 =A CB +L A W 2 +W 3 R B, V 1 = R A2 (C 2 R A1 C 1 B 1 B 2+ U 1 B 2 )D 1 +A 2W 4 +W 5 R D1, and W 1, W 2,, W 6 are arbitrary matrices over C with appropriate sizes. Lemma 3 18 C m1 k, and P C l n1 be given. Then, Let A C m n, B C m k, C C l n, D C m p, E C q n, Q (i) r(a) + r(r A B) = r(b) + r(r B A) = r A B ; A (ii) r(a) + r(cl A ) = r(c) + r(al C ) = r ; C A B (iii) r(b) + r(c) + r(r B AL C ) = r. C 0 Lemma 4 27 Let f(x, Y ) = A BXD CY F be a matrix expression over C. Denote A A C A C A B C k min = r A B C + r D + max r r r D 0, D 0 D 0 0 F F 0 A B } A B A B C r r r D 0, F 0 F 0 0 F 0 and k max = min r A B A A B C, r D,r F 0 F, r } A C. D 0 Then, k min rf(x, Y ) k max. 2 Some solvability conditions and general solution to system (3) In this section, we derive some solvability conditions and the general solution to the system of mixed Sylvester matrix equations (3). Now, we give the fundamental theorem of this paper. Theorem 1 Let A i, B i, C j, D j, and E j (i = 1, 2, 3, j = 1, 2) be given. Set A 4 = C 1 L A1, B 4 = R A2 D 1, C 4 = E 1 C 1 A 1 B 1 + B 2 A 2 D 1, A 5 = C 2 L A3, B 5 = R A2 D 2, C 5 = E 2 C 2 A 3 B 3 + B 2 A 2 D 2, A = R A5 A 4, B = B 5 L (RB4 B 5), C = R A5 (R A4 C 4 B 4 B 5 C 5 )L (RB4 B 5).

5 406 Communication on Applied Mathematics and Computation Vol. 28 Then, the following statements are equivalent. (i) The system of mixed Sylvester matrix equations (3) is consistent; (ii) R Ak B k = 0 (k = 1, 3), B 2 L A2 = 0, R A4 C 4 L B4 = 0, CL B = 0, R A C = 0; (iii) r A k B k = r(a k ) (k = 1, 3), r B2 A 2 = r(a 2 ), E 1 C 1 B 2 r D 1 0 A 2 = r B 1 0 E 2 C 2 B 2 r D 2 0 A 2 = r C2 + r A 2 D 1, B 3 A 3 0 C 1 C 2 E 1 E 2 B D 1 D 2 A C 1 C 2 2 r 0 B = r 0 + r A 2 D 1 D 2. 0 B 3 0 A 3 + r A 2 D 2, In this case, the general solution to the system of mixed Sylvester matrix equations (3) can be expressed as where X = A 1 B 1 + L A1 U 1, Y = B 2 A 2 + U 2R A2, Z = A 3 B 3 + L A3 U 3, U 1 = A 4 C 4 + V 1 B 4 + L A4 W 1, U 2 = R A4 C 4 B 4 + A 4V 1 + V 2 R B4, (5) U 3 = A 5 (C 5 R A4 C 4 B 4 B 5 + A 4 V 1 B 5 ) + W 4 R B4 B 5 + L A5 W 6, (6) V 1 = A CB + L A W 2 + W 3 R B, V 2 = R A5 (C 5 R A4 C 4 B 4 B 5 + A 4 V 1 B 5 )(R B4 B 5 ) + A 5 W 4 + W 5 R (RB4 B 5), and W 1, W 2,, W 6 are arbitrary matrices over C with appropriate sizes. Proof (i) (ii): We separate the equations in the system of mixed Sylvester matrix equations (3) into two groups X = B 1, Y A 2 = B 2, A 3 Z = B 3, (7) C 1 X Y D 1 = E 1, C 2 Z Y D 2 = E 2. (8)

6 No. 4 XU Huan, et al.: Solutions of mixed generalized Sylvester matrix equations 407 It follows from Lemma 1 that matrix equations in (7) are consistent, respectively, if and only if R A1 B 1 = 0, B 2 L A2 = 0, R A3 B 3 = 0. The general solutions to these matrix equations in (7) can be expressed as X = A 1 B 1 + L A1 U 1, Y = B 2 A 2 + U 2R A2, Z = A 3 B 3 + L A3 U 3, (9) where U 1, U 2, and U 3 are arbitrary. Substituting (9) into (8) gives L A1 U 1 U 2 R A2 D 1 = E 1 C 1 A 1 B 1 + B 2 A 2 D 1, C 2 L A3 U 3 U 2 R A2 D 2 = E 2 C 2 A 3 B 3 + B 2 A 2 D 2, i.e., A4 U 1 U 2 B 4 = C 4, A 5 U 3 U 2 B 5 = C 5. (10) Hence, the system (3) is consistent if and only if the matrix equations in (7) and (10) are consistent, respectively. It follows from Lemma 2 that the system of matrix equations (10) is consistent if and only if R A4 C 4 L B4 = 0, CL B = 0, R A C = 0. We know by Lemma 2 that the general solutions of the system of matrix equations (10) can be expressed as (5) (6). (ii) (iii): It follows from Lemma 2 that R Ak B k = 0 r(r Ak B k ) = 0 r A k B k = r(a k )(k = 1, 3), B 2 L A2 = 0 r(b 2 L A2 ) = 0 r B2 A 2 = r(a 2 ). From Lemma 1, R Ak B k = 0 (k = 1, 3), B 2 L A2 = 0, there exist X 0, Y 0, and Z 0 such that X 0 = B 1, Y 0 A 2 = B 2, and A 3 Z 0 = B 3. Applying Lemma 3 to R A4 C 4 L B4 = 0 gives C4 A 4 R A4 C 4 L B4 = 0 r(r A4 C 4 L B4 ) = 0 r = r(a 4 ) + r(b 4 ) B 4 0 C 4 C 1 L A1 r R A2 D 1 0 C 4 C 1 0 =r(c 1 L A1 )+r(r A2 D 1 ) r D 1 0 A 2 =r 0 0 +r A 2 D 1

7 408 Communication on Applied Mathematics and Computation Vol. 28 E 1 C 1 X 0 + Y 0 D 1 C 1 0 r D 1 0 A 2 = r 0 0 E 1 C 1 B 2 r D 1 0 A 2 = r B r A 2 D 1 + r A 2 D 1. Similarly, we have C5 A 5 CL B = 0 r(cl B ) = 0 r = r(a 5 ) + r(b 5 ) B 5 0 C 5 C 2 L A3 r R A2 D 2 0 E 2 C 2 Z 0 + Y 0 D 2 C 2 0 r D 2 0 A 2 = r 0 E 2 C 2 B 2 r D 2 0 A 2 = r B 3 A 3 0 = r(c 2 L A3 ) + r(r A2 D 2 ) C2 A 3 C2 A 3 + r A 2 D 2 + r A 2 D 2, A4 R A4 C 4 B 4 R A C = 0 r(r A C) = 0 r B 5 C 5 A 5 = r A 4 A 5 + r(r B4 B 5 ) 0 R B4 B 5 0 A4 A 5 C 5 C 4 r = r A 4 A 5 + r B 4 B B 5 B 4 C 1 C 2 E 1 E 2 B D 1 D 2 A C 1 C 2 2 r 0 B = r 0 + r A 2 D 1 D 2. 0 B 3 0 (i)= (iii): Assume that the system of mixed Sylvester matrix equations (3) has a solution, say, (X, Y, Z). Then, we have r B 1 = r X = r( ), ra 3 B 3 = ra 3 A 3 Z = r(a 3 ), B2 Y A2 r = r = r(a 2 ), A 2 A 2

8 No. 4 XU Huan, et al.: Solutions of mixed generalized Sylvester matrix equations 409 E 1 C 1 B 2 C 1 X Y D 1 C 1 Y A 2 r D 1 0 A 2 = r D 1 0 A 2 = r B 1 0 X 0 E 2 C 2 B 2 C 2 Z Y D 2 C 2 Y A 2 r D 2 0 A 2 = r D 2 0 A 2 = r B 3 A 3 0 A 3 Z A 3 0 C2 A 3 + r A 2 D 1, + r A 2 D 2, C 1 C 2 E 1 E 2 B 2 C 1 C 2 C 1 X Y D 1 C 2 Z Y D 2 Y A D 1 D 2 A 2 r 0 B = r 0 0 D 1 D 2 A 2 0 X B 3 0 Z 0 C 1 C 2 = r 0 + r A 2 D 1 D 2. In Theorem 1, assuming A 2 and B 2 vanish, we obtain the general solution to the system A1 X = B 1, A 3 Z = B 3, (11) C 1 X Y D 1 = E 1, C 2 Z Y D 2 = E 2. Corollary 1 Let A i, B i, C j, D j, and E j (i = 1, 3, j = 1, 2) be given. Set A 4 = C 1 L A1, C 4 = E 1 C 1 A 1 B 1, A 5 = C 2 L A3, C 5 = E 2 C 2 A 3 B 3, A = R A5 A 4, B = B 2 L (RD1 D 2), C = R A5 (R A4 C 4 D 1 D 2 C 5 )L (RD1 D 2). Then, the following statements are equivalent. (i) The system (11) is consistent; (ii) (iii) R Ak B k = 0 (k = 1, 3), R A4 C 4 L D1 = 0, CL B = 0, R A C = 0; E 1 C 1 r D 1 0 = r B 1 r A k B k = r(a k )(k = 1, 3), E 2 C 2 + r(d 1 ), r D 2 0 = r B 3 A 3 C2 C 1 C 2 E 1 E D 1 D C 1 C 2 2 r 0 B 1 0 = r 0 + r D 1 D 2. 0 B 3 A 3 + r(d 2 ),

9 410 Communication on Applied Mathematics and Computation Vol. 28 In this case, the general solution to the system (11) can be expressed as X = A 1 B 1 + L A1 U, Y = R A4 C 4 D 1 + A 4U 1 + V 1 R D1, Z = A 3 B 3 + L A3 V, where U = A 4 C 4 + U 1 D 1 + L A4 W 1, V = A 5 (C 5 R A4 C 4 D 1 D 2 + A 4 U 1 D 2 ) + W 4 R D1 D 2 + L A5 W 6, U 1 = A CB + L A W 2 + W 3 R B, V 1 = R A5 (C 5 R A4 C 4 D 1 D 2 + A 4 U 1 D 2 )(R D1 D 2 ) + A 5 W 4 + W 5 R (RD1 D 2), and W 1, W 2,, W 6 are arbitrary matrices over C with appropriate sizes. 3 Admissible ranks of general solution of system (3) We now turn our attention to the admissible ranks of the general solution of (3). For convenience, we use the following notations: } J 1 = X C m n X = B 1, Y A 2 = B 2, A 3 Z = B 3,, C 1 X Y D 1 = E 1, C 2 Z Y D 2 = E 2 J 2 = Z C p q X = B 1, Y A 2 = B 2, A 3 Z = B 3, C 1 X Y D 1 = E 1, C 2 Z Y D 2 = E 2 Theorem 2 Let A i, B i, C j, D j, E j (i = 1, 2,, 5, j = 1, 2), A, B, and C be defined as in Theorem 1. Suppose the system of mixed Sylvester matrix equations (3) is consistent. Then, }. x min r(x) X J 1 x max, z min r(z) Z J 2 z max, where D 1 A 2 x max = min n, m + r(b 1) r( ), m + r E 1 B 2 r r(a 2 ), B 1 0 D 1 D 2 0 A 2 E 1 E 2 C 2 0 C 1 C 2 m+r B r 0 r A 2 D 2, (12) 0 B 3 0 0

10 No. 4 XU Huan, et al.: Solutions of mixed generalized Sylvester matrix equations 411 D 1 A 2 x min = r(b 1 )+r E 1 B 2 B 1 0 D 1 D 2 0 A 2 D 1 A 2 0 E 1 E 2 C 2 0 +r B r E 1 B 2 C 2 B r D1 D 2 A 2 B B D 2 A 2 z max = min q, p + r(b C2 3) r(a 3 ), p + r E 2 B 2 r r(a 2 ), A 3 B 3 0 D 2 D 1 0 A 2 E 2 E 1 C 1 0 C 1 C 2 p + r B r 0 r A 2 D 1, 0 B D 2 A 2 z min = r(b 3 ) + r E 2 B 2 B 3 0 D 2 D 1 0 A 2 D 2 A 2 0 E 2 E 1 C 1 0 +r B r E 2 B 2 C 1 B r D2 D 1 A 2 B B , (13) Proof By Theorem 1, the expression of X in the system of mixed Sylvester matrix equations (3) can be expressed as X = X 0 + L A1 L A4 W 1 + L A1 L A W 2 B 4 + L A1 W 3 R B B 4, (14). where X 0 is a special solution of (3), W 1, W 2, and W 3 are arbitrary matrices over C with appropriate sizes. Note that R(L A4 ) R(L A ). It follows from Lemma 4 that r X 0 X0 L A1 L A4 X0 L A1 L A L A1 + r + r B 4 0 R B B 4 0 X0 L A1 L A r B 4 0 r(x) X J 1 min r X0 L A1 n, r X 0 L A1,r r(l A1 L A4 ) R B B 4 0 X0 L A1 L A4 B 4 0, r X0 L A1 L A R B B 4 0 }. (15)

11 412 Communication on Applied Mathematics and Computation Vol. 28 Applying Lemma 3, using the block Gaussian elimination and X 0 = B 1, Y 0 A 2 = B 2, A 3 Z 0 = B 3, C 1 X 0 Y 0 D 1 = E 1, C 2 Z 0 Y 0 D 2 = E 2, we have r X 0 L A1 = m + r(b 1 ) r( ), r(l A1 L A4 ) = m r X0 L A1 L A4 r B 4 0 D 1 A 2 = m + r E 1 B 2 r B 1 0, (16) r(a 2 ), (17) D 1 D 2 0 A 2 X0 L A1 L A E 1 E 2 C 2 0 C 1 C 2 r = m + r R B B 4 0 B r 0 r A 2 D 2, (18) 0 B D 1 A 2 0 X0 L A1 L A E 1 B 2 C C 1 C 2 2 r = m + r B 4 0 B r(a 2) r 0, (19) 0 X0 L A1 r = m + r R B B 4 0 D1 D 2 A 2 B r( ) r A 2 D 2. (20) Substituting (16) (20) into (15) yields (12) and (13). Using A 3, B 3, C 2, D 2, and E 2 instead of, B 1, C 1, D 1, and E 1 in (12) and (13), we can obtain the admissible ranks of Z in (3). In Theorem 2, assuming A 2 and B 2 vanish, we obtain the admissible ranks of X and Z in the system (11). Corollary 2 Let A i, B i, C j, D j, and E j (i = 1, 3, j = 1, 2) be given. Suppose the system (11) is consistent. We take two matrix sets as Then, S 1 = S 2 = X C m n X = B 1, A 3 Z = B 3, C 1 X Y D 1 = E 1, C 2 Z Y D 2 = E 2 Z C p q X = B 1, A 3 Z = B 3, C 1 X Y D 1 = E 1, C 2 Z Y D 2 = E 2 } }., x min r(x) X S 1 x max, z min r(z) Z S 2 z max,

12 No. 4 XU Huan, et al.: Solutions of mixed generalized Sylvester matrix equations 413 where x max =min n, m + r(b 1) r( ), m + r D 1 E 1 B 1 r, D 1 D 2 0 E 1 E 2 C C 1 C 2 2 m+r B r 0 r(d 2 ), 0 B 3 0 x min = r(b 1 ) + r D 1 E 1 B 1 D 1 D 2 0 D 1 0 E 1 E 2 C 2 + r B r E 1 C 2 B 1 0 r D1 D 2 B B 3 0 z max =min q, p + r(b 3) r(a 3 ), p + r z min = r(b 3 ) + r D 2 E 2 B 3 D 2 E 2 B 3 r C2 A 3, D 2 D 1 0 E 2 E 1 C C 1 C 2 1 p+r B r 0 r(d 1 ), 0 B 1 0 D 2 D 1 0 D 2 0 E 2 E 1 C 1 + r B r E 2 C 1 B 3 0 r D2 D 1 B B Corollary 3 Let A i, B i, and C i (i = 1, 2) be given. Suppose that the mixed pairs of Sylvester matrix equations (1) are consistent. Then, x min r(x) x max, ẑ min r(z) ẑ max, X Y B 1=C 1,A 2Z Y B 2=C 2 X Y B 1=C 1,A 2Z Y B 2=C 2 x max = min m, n, m + r B1 C 1 r A 2 r(b 2 ) B1 B 2 0 r( ), m + r C 1 C 2 A 2 },,. x min = r B1 C 1 B1 B r r B 1 C 1 C 2 A 2 B1 0 B 2 r, C 1 A 2

13 414 Communication on Applied Mathematics and Computation Vol. 28 ẑ max = min p, q, p + r B2 C 2 r A 2 r(b 1 ) B1 B 2 0 r(a 2 ), p + r C 1 C 2 }, ẑ min = r B2 C 2 B1 B r r B 1 C 1 C 2 B2 0 B 2 r. C 2 Remark 1 Corollary 3 is the main result of Conclusions We have derived some solvability conditions for the existence of the general solution to the system of mixed generalized pairs of Sylvester matrix equations (3). Based on the expression of the solution to (3), we have presented the the admissible ranks of the general solution of system (3). This paper extends the major results of 23. References 1 Zhang Y, Jiang D, Wang J. A recurrent neural network for solving Sylvester equation with time-varying coefficients J. IEEE Transactions on Neural Networks, 2002, 13(5): Iii R K, Bhattacharyya S P. Robust and well conditioned eigenstructure assignment via Sylvester s equation J. Optimal Control Applications and Methods, 1983, 4(3): Duan G R. Eigenstructure assignment and response analysis in descriptor linear systems with state feedback control J. International Journal of Control, 1998, 69(5): Syrmos V L, Lewis F L. Output feedback eigenstructure assignment using two Sylvester equations J. IEEE Transations on Automomic Control, 1993, 38(3): Duan G R. Solution to matrix equation AV + BW = EV F and eigenstructure assignment for descriptor systems J. Automatica, 1992, 28(3): Fletcher L R, Kautsky J, Nichols N K. Eigenstructure assignment in descriptor systems J. IEEE Transactions on Automatic Control, 1986, 31(12): Baksalary J K, Kala P. The matrix equation AX + Y B = C J. Linear Algebra Appl, 1979, 25: Bao L, Lin Y Q, Wei Y M. A new projection method for solving large Sylvester equations J. Appl Num Math, 2007, 57: Calvetti D, Lewis B, Reichel L. On the solution of large Sylvester-observer equation J. Numer Linear Algebra Appl, 2001, 8: Chen T, Francis B A. Optimal Sampled-data Control Systems M. London: Springer-Verlag, Castelan E B, da Silva V Gomes. On the solution of a Sylvester matrix equation appearing in descriptor systems control theory J. Syst Contr Lett, 2005, 54: Dehghan M, Hajarian M. Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations J. Appl Math Model, 2011, 35: Ding F, Chen T, Iterative least squares solutions of coupled Sylvester matrix equations J. Syst Contr Lett, 2005, 54: El Guennouni A, Jbilou K, Riquet J. Block Krylov subspace methods for solving large Sylvester equations J. Numer Algorithms, 2002, 29: Kägström B. A perturbation analysis of the generalized Sylvester equation (AR LB, DR LE) = (C,F) J. SIAM J Matrix Anal Appl, 1994, 15:

14 No. 4 XU Huan, et al.: Solutions of mixed generalized Sylvester matrix equations Liu Y H. Ranks of solutions of the linear matrix equation AX + Y B = C J. Comput Math Appl, 2006, 52: Lee S G, Vu Q P. Simultaneous solutions of matrix equations and simultaneous equivalence of matrices J. Linear Algebra Appl, 2012, 437: Marsaglia G, Styan G P H. Equalities and inequalities for ranks of matrices J. Linear and Multilinear Algebra, 1974, 2: Penrose R. A generalized inverse for matrices J. Proc Camb Philos Soc, 1955, 51: Roth W E. The equation AX Y B = C and AX XB = C in matrices J. Proc Amer Math Soc, 1952, 3: Tsui C C. A complete analytical solution to the equation T A F T = LC and its applications J. IEEE Trans Autom Control, 1987, 8: Wang Q W, Sun J H, Li S Z. Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra J. Linear Algebra Appl, 2002, 353: Wang Q W, He Z H. Solvability conditions and general solution for mixed Sylvester equations J. Automatica, 2013, 49: Wimmer H K. Consistency of a pair of generalized Sylvester equations J. IEEE Trans Autom Control, 1994, 39: Zhou B, Li Z Y, Duan G R, Wang Y. Weighted least squares solutions to general coupled Sylvester matrix equations J. J Comput Appl Math, 2009, 224: He Z H, Wang Q W. A pair of mixed generalized Sylvester matrix equations J. Journal of Shanghai University: Natural Science, 2014, 20(2). DOI: /j.issn Chu D L, Hung Y S, Woerdeman H J. Inertia and rank characterizations of some matrix expressions J. SIAM J Matrix Anal Appl, 2009, 31:

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