Applied Mathematics and Computation

Applied Mathematics and Computation 212 (2009) 327 336 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Solutions to a family of matrix equations by using the Kronecker matrix polynomials Bin Zhou a, *, Zhao-Yan Li b, Guang-Ren Duan a, Yong Wang b a Center for Control Theory and Guidance Technology, Harbin Institute of Technology, P.O. Box 416, Harbin 150001, Heilongjiang, PR China b Department of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China article info abstract Keywords: Generalized Sylvester matrix equations Kronecker matrix polynomials Coprime Closed form solutions Linear system theory Closed form solutions to a family of generalized Sylvester matrix equation in form of P / A ixf i þ P w B kyf k ¼ P u E jrf j are given by using the so-called Kronecker matrix polynomials. It is found that the structure of the solutions is independent of the orders /; w and u. This type of uniform closed form solutions includes our early results as special cases. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in linear systems. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction When dealing with the analysis and design problems associated with the linear system _xðtþ ¼AxðtÞþBuðtÞ where A and B are known matrices with appropriate dimensions, a matrix equation in the form of AX XF ¼ BY þ R where F and R are two known matrices, and ðx; YÞ is a pair of matrices to be determined, are encountered. The homogeneous case of the above equation is AX XF ¼ BY. These two first-order generalized Sylvester matrix equations are widely used in some control problems such as pole and eigenstructure assignment design [15,16,24], Luenberger-type observer design [17,19], and robust fault detection [5,18]. When dealing with the analysis and design problems associated with the descriptor linear system E_x ¼ Ax þ Bu where A; E and B are known matrices having appropriate dimensions, we often encounter a matrix equation of the form AX EXF ¼ BY þ R where the matrices F and R are known with proper dimensions, and the matrix pair ðx; YÞ is to be determined. In the case of R ¼ 0, the above mentioned nonhomogeneous equation becomes AX EXF ¼ BY which is closely related with pole placement, eigenstructure assignment [1], observer design [6], and some other problems (see [10] for a detailed introduction). Many control problems, such as pole assignment [23 25] and eigenstructure assignment [4,26], for the second-order linear system M x þ D_x þ Kx ¼ Bu where M; D; K and B are known matrices with appropriate dimensions, are closely related with a matrix equation of the form MVF 2 þ DVF þ KV ¼ BW þ R where F and R are given matrices, and ðx; YÞ is a pair of matrices to be determined. This type of equation is called the second-order nonhomogeneous generalized Sylvester matrix equation. When R ¼ 0, it becomes the homogeneous one MVF 2 þ DVF þ KV ¼ BW. It can be shown that certain control problems, such as pole assignment, eigenstructure assignment and observer design [3], of the high-order linear system P m A ix ðiþ ¼ Bu where A i ; i ¼ 0; 1;...; m and B are known matrices, are closely related with * Corresponding author. E-mail addresses: binzhou@hit.edu.cn, binzhoulee@163.com (B. Zhou), zhaoyanlee@gmail.com (Z.-Y. Li). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.02.021

328 B. Zhou et al. / Applied Mathematics and Computation 212 (2009) 327 336 a matrix equation of the form [3] P m A ixf i ¼ BY þ R in which F and R are some known matrices having proper dimensions, and ðx; YÞ is a pair of matrices to be determined. Such a type of equation is called the mth order nonhomogeneous generalized Sylvester matrix equation [3]. Its homogeneous part is P m A ixf i ¼ BY. Obviously, the above mth order generalized Sylvester equation includes both the above mentioned first-order equations and second-order equations as special cases. A large amount of prior work has been devoted to solving the generalized nonhomogeneous and homogeneous Sylvester matrix equations given above with F in the Jordan or companion form. See [1,2,9,20,21]. However, the restriction that F is in Jordan canonical form is an obvious disadvantage. Because both the existing result itself and its proof in the literature (see, e.g., [1]) heavily rely on the concrete structure of F as a Jordan canonical form, one can not obtain the solutions of the firstorder generalized Sylvester matrix equation for general matrix F by using the existing result straightway. To deal with the general matrix F, one should transform F into its Jordan canonical form beforehand, and then transform the first-order generalized Sylvester matrix equations into an equivalent one, in which the Jordan form appears. The solutions obtained by using this method are very complicated for involving the computation of Jordan canonical form, and will not be convenient for use in practice. Therefore, it is useful and interesting to give some complete and explicit solution by using the general matrix F itself directly. For these reasons, the authors have done some work on this problem recently. In detail, we have proposed in [10] for the first time a complete parametric solutions to the equation AX EXF ¼ BW with F an arbitrary square matrix and then extended it to the second-order case in [8]. As a result, it is found that solutions to equations AX EXF ¼ BW and MVF 2 þ DVF þ KV ¼ BW are closely related with some matrix polynomials associated with the coefficient matrices and have the same structure that is independent of the coefficient matrices and the order of the equation (see [8,10] for details). Therefore, one aim of this paper is to provide a methodology to unify the solutions to all these equations. The results are obtained by using a different method from that used in [8,10]. Another aim of this paper is to extend this family of generalized Sylvester matrix equation to a more general case that the unknown variables are more than two. It will be found that the result can also be given in a very similar way. The results presented here also generalize those in [11] in some aspects. Especially, the so-called Kronecker matrix polynomial given in this paper seems more convenient to use than the so-called generalized Sylvester mapping presented in that paper. Finally, we should point out that matrix equation is a very hot topic and many relating results can be found in the literature in recent years. For related references, see [13,14,22,27,28] and the references given there. The remainder of this paper is organized as follows. In Section 2, we introduce the so-called Kronecker matrix polynomials whose properties are studied. By using the Kronecker matrix polynomials, closed form solutions to a family of generalized Sylvester matrix equations are presented in Section 3. Some further extensions are given in Section 4. Section 5 concludes the paper. Notations: In this paper, we use to denote the Kronecker product. For an m n matrix R ¼½r ij Š, the so-called column stretching function csðþ and the row stretching function rsðþ are respectively defined as csðrþ ¼½r 11 r m1 r 1n r mn Š T ; rsðrþ ¼½r 11 r 1n r m1 r mn Š: For three matrices M; X and N with appropriate dimensions, we have the following well-known results related with the stretching operations: csðmxnþ ¼ðN T MÞcsðXÞ; rsðmxnþ ¼rsðXÞðM T NÞ: ð1þ ð2þ Furthermore, we use rðaþ to denote the eigenvalue set of matrix A, col½a 1 ; A 2 ;...; A s Š¼col½A i Š s i¼1 to denote a block column matrix with the ith matrix A i, and diag½d i Š s i¼1 to denote a diagonal matrix with the ith element being d i, respectively. 2. The Kronecker matrix polynomials For a matrix polynomial AðsÞ ¼ Xx A i s i ; A i 2 R nq ; i ¼ 0; 1;...; x; where x P 0 is a given integer, and a square matrix F 2 R pp. we define the so-called Kronecker matrix polynomial as A ðfþ ¼ Xx ðf T Þ i A i : Obviously, A ðfþ is produced from AðsÞ by the substitution ðs; Þ $ ðf T ; Þ. This kind of matrix polynomial was initially proposed in our early work [10] (see Eq. (11) and Eq. (17) in [10]) to study the matrix equation (13) given later and some of its properties have been reported there (Lemma 1 in [10]). A different definition for a similar kind of matrix polynomials from the mapping point of view was first given in [30]. The Kronecker matrix polynomial has the following properties shown in Lemmas 1 3 whose proofs are simple and omitted here.

B. Zhou et al. / Applied Mathematics and Computation 212 (2009) 327 336 329 Lemma 1. Let AðsÞ 2R nq ½sŠ; BðsÞ 2R qr ½sŠ and AðsÞBðsÞ ¼CðsÞ. Then C ðfþ ¼A ðfþb ðfþ: Let AðsÞ; BðsÞ 2R nq ½sŠ and AðsÞþBðsÞ ¼DðsÞ. Then ¼A ðfþþb ðfþ: ð3þ ð4þ Lemma 2. Let A k ðsþ 2R nq k ½sŠ; B k ðsþ 2R q k r ½sŠ; k ¼ 1; 2;...; m, and Then X m k¼1 X m k¼1 A k ðsþb k ðsþ ¼CðsÞ: ða k Þ ðfþðb k Þ ðfþ ¼C ðfþ: It follows from Lemma 1 that if AðsÞ 2R qq ½sŠ and BðsÞ 2R qq ½sŠ satisfy AðsÞBðsÞ ¼I q, i.e., AðsÞ (or BðsÞ) is invertible, then A ðfþb ðfþ ¼I q F 0 ¼ I qp which implies that A ðfþ (or B ðfþ) is also invertible. Using this fact, we can get the following result. Lemma 3. Let AðsÞ 2R nq ½sŠ be an arbitrary matrix polynomial, PðsÞ 2R nn ½sŠ and QðsÞ 2R qq ½sŠ be two unimodular matrix polynomials and satisfy the Smith normal factorization PðsÞAðsÞQðsÞ ¼ DðsÞ 0 ; 0 0 where DðsÞ ¼diag½d i ðsþš r i¼1, and d 1ðsÞjjd r ðsþ. Then rankða ðfþþ ¼ rankðþ ¼ Xr i¼1 rankðd i ðfþþ: The following definition is directly adopted from [10]. Definition 1. Let F 2 R pp be an arbitrary matrix. Then 1. A pair of polynomial matrices NðsÞ 2R nr ½sŠ and DðsÞ 2R qr ½sŠ is said to be F-right coprime if rank NðkÞ ¼ r; 8k 2 rðfþ: ð5þ DðkÞ 2. A pair of polynomial matrices HðsÞ 2R mn ½sŠ and LðsÞ 2R mq ½sŠ is said to be F-left coprime if rank½hðkþ LðkÞ Š ¼ m; 8k 2 rðfþ: ð6þ Remark 1. The definition of the F-right coprimeness (F-left coprimeness) of two matrix polynomials NðsÞ and DðsÞ can be easily extended to a series of matrix polynomials having the same columns (rows) and a single matrix polynomial. The following proposition was first established in [10]. But the proof given there is very complicated. Here a new simple and alterative proof is given by using Lemma 3. Proposition 1. Let F 2 R pp be an arbitrary matrix. Then 1. A pair of polynomial matrices NðsÞ 2R nr ½sŠ and DðsÞ 2R qr ½sŠ is F-right coprime if and only if rank N ðfþ ¼ rank½w ðfþš ¼ rp; where WðsÞ ¼col½NðsÞ; DðsÞŠ; 2. A pair of polynomial matrices HðsÞ 2R mn ½sŠ and LðsÞ 2R mq ½sŠ is F-left coprime if and only if rank½h ðfþ L ðfþ Š ¼ rank½m ðfþš ¼ mp; ð7þ where MðsÞ ¼½HðsÞ; LðsÞŠ. Proof. We only show the first item since the second item can be shown similarly. First, the relation rank N ðfþ ¼ rank½w ðfþš

330 B. Zhou et al. / Applied Mathematics and Computation 212 (2009) 327 336 can be easily obtained by changing some rows of the associated matrix. See [10]. Second, we show rank½w ðfþš ¼ rp. Let TðsÞ ¼diag½t i ðsþš / i¼1 2 R// ½sŠ; / 6 r be the Smith canonical form of the matrix polynomial WðsÞ, i.e., there exist two unimodular polynomial matrices PðsÞ and QðsÞ such that PðsÞWðsÞQðsÞ ¼ TðsÞ 0 0 0 According to Lemma 3, rank½w ðfþš ¼ rp is equivalent to : ð8þ rank½t ðfþš ¼ X/ rankðt i ðfþþ ¼ rp: i¼1 Furthermore, according to (8) and (5), NðsÞ and DðsÞ are F-right coprime if and only if ð9þ rankðwðkþþ ¼ X/ rankðt i ðkþþ ¼ r; 8k 2 rðfþ; ð10þ i¼1 which implies that / ¼ r. The equivalence between (9) and (10) can be easily observed since rðt i ðfþþ ¼ ft i ðkþjk 2 rðfþg; i ¼ 1; 2;...; r: This completes the proof. 3. Solutions to a family of matrix equations 3.1. The homogeneous generalized Sylvester matrix equations In this subsection, we consider the following general homogeneous generalized Sylvester matrix equation X / A i XF i þ Xw B k YF k ¼ 0; where A i 2 R nn ; i ¼ 0; 1;...;/ and B i 2 R nr ; i ¼ 0; 1;...; w; F 2 C pp are some known matrices and X 2 R np ; Y 2 R rp are matrices to be determined. Clearly, this equation includes the following equations mentioned in Section 1 as special cases: AX XF ¼ BY; ð12þ AX EXF ¼ BY; ð13þ MXF 2 þ DXF þ KX ¼ BY; ð14þ X m A i XF i ¼ BY: Define a pair of polynomial matrices AðsÞ and BðsÞ as follows: AðsÞ ¼ X/ A i s i 2 R nn ½sŠ; BðsÞ ¼ Xw B i s i 2 R nr ½sŠ: Putting csðþ on both sides of (11) and using (2) gives csðxþ ½A ðfþ B ðfþ Š ¼ 0: csðyþ Let S and fs be the sets of the solutions of (11) and (17), respectively, i.e., ( ) X S ¼ : X/ A i XF i þ Xw B k YF k ¼ 0 ; Y and fs ¼ fw : ½A ðfþ B ðfþ Šw ¼ 0g: ð18þ Then the map s : S! fs defined by X s : # csðxþ ; ð19þ Y csðyþ is a linear isomorphism. Therefore, we need only to consider the Eq. (17). ð11þ ð15þ ð16þ ð17þ

B. Zhou et al. / Applied Mathematics and Computation 212 (2009) 327 336 331 Proposition 2. Let fs be given by (18). Then dimðfsþ ¼rp if and only if AðsÞ and BðsÞ are F-left coprime. Proof. Obviously, dimðfsþ ¼rp if and only if rank½a ðfþ B ðfþ Š ¼ np: ð20þ According to Proposition 1, Eq. (20) holds if and only if AðsÞ and BðsÞ are F-left coprime. This completes the proof. Assume that there exists a pair of matrix polynomials NðsÞ ¼ P x N is i 2 R nr ½sŠ and DðsÞ ¼ P x D is i 2 R rr ½sŠ, where x P 0 is some integer, such that AðsÞNðsÞþBðsÞDðsÞ ¼0: ð21þ Theorem 1. Suppose that AðsÞ and BðsÞ are F -left coprime and NðsÞ and DðsÞ are F-right coprime. Then N ðfþ fs ¼ csðzþ : Z 2 C rp Proof. First, since NðsÞ and DðsÞ satisfy (21), by using Lemma 2 we have N ðfþ A ðfþn ðfþþb ðfþ ¼½A ðfþ B ðfþ Š ¼ 0; which implies that w ¼ N ðfþ cs ðzþ; Z 2 C rp ; : ð22þ satisfies the Eq. (17). Second, since dimðzþ ¼rp, the relation dimðfsþ ¼rp holds if and only if rank N ðfþ ¼ rp: According to Proposition 1, the above equation is equivalent to the F-coprimeness of the matrix polynomials NðsÞ and DðsÞ. The proof is completed. In view of (19), the equation in (1) and Theorem 1, we clearly have the following corollary. Corollary 1. The solution space of (11) can be explicitly expressed by 82 P x 3 9 >< N i ZF i >= S ¼ 6 4 P x 7 D i ZF i 5 : Z 2 Crp : ð23þ >: >; Remark 2. The expression (23) was initially established in [10] for the first-order Eq. (13) and then extended to secondorder case (14) in [7,8]. It follows from Corollary 1 that this type of solutions also holds for the more general case (11). That is to say, the structure of the solution is independent of the order of the equation. 3.2. The nonhomogeneous generalized Sylvester matrix equations We now consider the nonhomogeneous extension of (11), i.e., a linear equation in the form of X / A i XF i þ Xw B k YF k ¼ Xu E j RF j ; where AðsÞ 2R nn ½sŠ and BðsÞ 2R nr ½sŠ are given by (16), EðsÞ ¼ Xu E j s j 2 R nq ½sŠ; ð24þ ð25þ is a given matrix polynomial, R 2 C qp and F 2 C pp are known matrices, and X 2 R np ; Y 2 R rp are matrices to be determined. Solution to a matrix equation with the right-hand-side similar to that of Eq. (24) was discussed in [29]. Clearly, the following matrix equations mentioned in Section 1 are special cases of (24):

332 B. Zhou et al. / Applied Mathematics and Computation 212 (2009) 327 336 AX XF ¼ BY þ R; AX EXF ¼ BY þ R; MXF 2 þ DXF þ KX ¼ BY þ R; X m A i XF i ¼ BY þ R: Let R be the set of the solutions to (24), that is, ( ) X R ¼ : X/ A i XF i þ Xw B k YF k ¼ Xu E j RF j : Y Similarly, Eq. (24) can be equivalently rewritten as csðxþ ½A ðfþ B ðfþ Š ¼ E ðfþcsðrþ: csðyþ Also, we denote ð26þ ð27þ ð28þ ð29þ ð30þ er ¼ ft : ½A ðfþ B ðfþ Št ¼ E ðfþcsðrþg: Assume that there exists a pair of matrix polynomials UðsÞ ¼ P u U is i 2 R nq ½sŠ and VðsÞ ¼ P u V is i 2 R rq ½sŠ, where u P 0is an integer, such that AðsÞUðsÞþBðsÞVðsÞ ¼EðsÞ: ð31þ Lemma 4. Let UðsÞ and VðsÞ satisfy (31). Then er 3 t s ¼ U ðfþ csðrþ: V ðfþ ð32þ Proof. Since UðsÞ and VðsÞ satisfy (31), using Lemma 2, we have U ðfþ ½A ðfþ B ðfþ Š csðrþ ¼ðA ðfþu ðfþþb ðfþv ðfþþcsðrþ ¼E ðfþcsðrþ; V ðfþ which implies that t s given in (32) satisfies the Eq. (30). This ends the proof. It follows from Lemma 4 that t s is a special solution to the nonhomogeneous Eq. (24). Combining Theorem 1 and Lemma 4 produces the following theorem. Theorem 2. Let AðsÞ and BðsÞ be F-left coprime, NðsÞ and DðsÞ satisfying (21) be F-right coprime, and UðsÞ and VðsÞ satisfy (31). Then N ðfþ U ðfþ csðzþ er ¼ : Z 2 C rp : V ðfþ csðrþ Similar to Corollary 1, we have the following result. Corollary 2. Let AðsÞ; BðsÞ; NðsÞ; DðsÞ; UðsÞ and VðsÞ be stated as in Theorem 2. Then 82 3 9 Px >< N i ZF i þ Pu U i RF i >= R ¼ 6 P x 7 4 D i ZF i þ Pu V i RF i 5 : Z 2 Crp : >: >; It follows from Theorems 1 and 2 that the pair of matrix polynomials ðnðsþ; DðsÞÞ satisfying (21) and the pair of matrix polynomials ðuðsþ; VðsÞÞ satisfying (31) are crucial for solving the Eqs. (11) and (24). The following result gives a method to determine these four matrix polynomials. A similar result can be found in [11]. Proposition 3. Assume that the matrix polynomials AðsÞ 2R nn ½sŠ and BðsÞ 2R nr ½sŠ satisfy rank½aðsþ BðsÞ Š¼n; 8s 2 C: ð33þ Then there exists two unimodular matrices PðsÞ and QðsÞ with appropriate dimensions satisfying PðsÞ½AðsÞ BðsÞ ŠQðsÞ ¼½0 I n Š:

B. Zhou et al. / Applied Mathematics and Computation 212 (2009) 327 336 333 Denote QðsÞ ¼ Q 11ðsÞ Q 12 ðsþ ; ð34þ Q 21 ðsþ Q 22 ðsþ where Q 21 ðsþ 2R rr ½sŠ. Then the polynomial matrices NðsÞ; DðsÞ; UðsÞ and VðsÞ satisfying (21) and (31) can be chosen as NðsÞ ¼Q 11 ðsþ; UðsÞ ¼Q 12 ðsþpðsþeðsþ; DðsÞ ¼Q 21 ðsþ; VðsÞ ¼Q 22 ðsþpðsþeðsþ: ð35þ Furthermore, the polynomial matrices NðsÞ and DðsÞ are right coprime (thus F-right coprime for arbitrary F). Remark 3. The rank condition (33) has a systematic meaning when the generalized Sylvester matrix Eq. (11) reduces to the special cases as (12) (15). For instance, condition (33) reduces to rank½si A BŠ ¼ n; 8s 2 C ð36þ for Eqs. (12) and (26), reduces to rank½se A BŠ ¼ n; 8s 2 C ð37þ for Eqs. (13) and (27), and reduces to rank Ms 2 þ Ds þ K B ¼ n; 8s 2 C ð38þ for Eqs. (14) and (28). Obviously, in these cases, conditions (36) (38) represent the controllability property of the corresponding linear systems [7,8]. Remark 4. Theorem 2 also generalizes the results in [11] where the right hand side of (24) is a simple constant matrix R. 4. Some extensions 4.1. The generalized Sylvester-observer matrix equation The generalized Sylvester-observer matrix equation has the form X / F i XA i þ Xw F k YC k ¼ 0; where A i 2 R nn ; i ¼ 0; 1;...; /; C k 2 R mn ; k ¼ 0; 1;...; w; F 2 C pp are known matrices and X 2 R pn ; Y 2 R pm are matrices to be determined. Similar to (16), we denote AðsÞ ¼ X/ A i s i 2 R nn ½sŠ; CðsÞ ¼ Xw C i s i 2 R mn ½sŠ: Such a matrix Eq. (39) is the dual form of (11) and has wide applications in observer design problems [12]. Obviously, Eq. (39) includes several equations, such as TA FT ¼ LC with T and L unknown studied in [20,21], the equation F 2 TM þ FTD þ TK þ FLR þ LQ ¼ 0, with T and L unknown studied in [12], as special cases. Also we denote ( ) F i XA i þ Xw F k YC k ¼ 0 : O ¼ ½X YŠ : X/ By using (2), Eq. (39) can be equivalently rewritten as A ðfþ ½rsðXÞ rsðyþ Š ¼ 0; C ðfþ which is clearly the dual form of (17). Theorem 3. Let AðsÞ and CðsÞ be F-right coprime, HðsÞ ¼ P - H is i 2 R mn ½sŠ and LðsÞ ¼ P - L is i 2 R mm ½sŠ where - P 0 is an integer, satisfying HðsÞAðsÞþLðsÞCðsÞ ¼0; be F-left coprime. Then P- O ¼ F i P- ZH i F i ZL i : Z 2 C pm : ð39þ

334 B. Zhou et al. / Applied Mathematics and Computation 212 (2009) 327 336 Remark 5. The following equation X / F i XA i þ Xw F k YC k ¼ Xu F j RE j is the dual form of (24). Thus corresponding results to Theorem 2 and Corollary 1 can be easily obtained. 4.2. Matrix equations with more than two unknowns We now consider a more general case in the following form where X h X x k k¼1 A ki X k F i ¼ 0; ð40þ A k ðsþ ¼ Xx k A ki s i 2 R nr k ; r k P 1; k ¼ 1; 2;...; h ð41þ i¼1 are some given matrix polynomials and F 2 R pp is a given square matrix. For instance, when h ¼ 2 and A 1 ðsþ ¼AðsÞ; A 2 ðsþ ¼ B, then (40) reduces to (15). The set of solutions to (40) is denoted by ( S h ¼ col½x i Š h i¼1 : Xh X x ) k A ki X k F i ¼ 0 : k¼1 Assume that A k ðsþ; k ¼ 1; 2;...; h given by (41) are F-left coprime, i.e., rank½a 1 ðkþ A 2 ðkþ A h ðkþ Š ¼ n; 8k 2 rðfþ; and there exists a series of matrix polynomials B i ðsþ 2R n k l ; l ¼ P h r i n; k ¼ 1; 2;...; h such that X h k¼1 A i ðsþb i ðsþ ¼0: Theorem 4. If the matrix polinomials B k ðsþ ¼ P wk B kis i ; k ¼ 1; 2;...; h are F-right coprime, then ( ) S h ¼ col½x i Š h i¼1 : X i ¼ Xwi B ik ZF k ; Z 2 C lp : We can consider solutions to the following family of generalized Sylvester matrix equation with more umknowns: X h X x k k¼1 A ki X k F i ¼ Xu E j RF j ; ð42þ where A k ðsþ; k ¼ 1; 2;...; h are given by (41), E(s) is given by (25) and R is a given matrix. In fact, the equation in (42) can be written as the following simple and unified form X x A i XF i ¼ Xu E j RF j ; i¼1 ð43þ where X ¼ col½x i Š h i¼1 and AðsÞ ¼½A 1 ðsþ A 2 ðsþ...a h ðsþš, Xx A i s i 2 R nr ; with x ¼ max 16k6h fx k g; r ¼ P h r k and A i ðsþ; i ¼ 1; 2;...; h given by (41). Then by using a similar technique used in Section 3.2, we can propose the following result without a proof. Theorem 5. Assume that AðsÞ is F-left coprime. Let NðsÞ 2R rðr nþ and MðsÞ 2R rq be such that AðsÞNðsÞ ¼0; AðsÞMðsÞ ¼EðsÞ: Moreover, assume that NðsÞ is F-right coprime and NðsÞ ¼ X - N i s i ; MðsÞ ¼ X p M i s i ;

B. Zhou et al. / Applied Mathematics and Computation 212 (2009) 327 336 335 where - and p are two integers. Then the solution space of Eq. (43) can be characterized as ( ) S A ¼ : X - N i ZF i þ Xp M i RF i : Z 2 C ðr nþp : 5. Conclusions This paper deals with a family of generalized Sylvester matrix equations in the form of (11) and (24), which include the first-order and second-order generalized Sylvester matrix Eqs. (12) (15) and (26) (29) as special cases, by using the Kronecker matrix polynomials. This paper has achieved the following: (A): The solutions to this class of equations are unified by using the so-called Kronecker matrix polynomials which gives a deep insight to the structure and the solution of this family of equations. The solutions also have a very simple, neat and elegant structure, and (B): All of the solutions to this family of generalized Sylvester matrix equations can be obtained without any assumption on matrix F and R as soon as some matrix polynomials associated with the equation are obtained. Acknowledgements The work of Bin Zhou and Guang-Ren Duan was partially supported by the Major Program of National Natural Science Foundation of China under Grant No. 60710002 and Program for Changjiang Scholars and Innovative Research Team in University. The work of Zhao-Yan Li and Yong Wang was supported by the National Natural Science Foundation of China (Grant No. 10771044) and the Natural Science Foundation of Heilongjiang Province (Grant No. 200605). The authors would like to thank the reviewers for their helpful comments which have helped them in improving the quality of this paper. References [1] G.R. 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