A Constraint System of Generalized Sylvester ...

A Constraint System of Generalized Sylvester Quaternion Matrix Equations

Abdur Rehman, Qing-Wen Wang, Ilyas Ali, Muhammad Akram & M. O. Ahmad

Advances in Applied Clifford Algebras ISSN 0188-7009 Adv. Appl. Clifford Algebras DOI 10.1007/s00006-017-0803-1

1 23

Your article is protected by copyright and all rights are held exclusively by Springer International Publishing AG. This e-offprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”.

1 23

Author's personal copy Adv. Appl. Clifford Algebras c 2017 Springer International Publishing AG  DOI 10.1007/s00006-017-0803-1

Advances in Applied Clifford Algebras

A Constraint System of Generalized Sylvester Quaternion Matrix Equations Abdur Rehman, Qing-Wen Wang ∗ , Ilyas Ali, Muhammad Akram and M. O. Ahmad Communicated by Rafal Ablamowicz Abstract. By keeping in mind the great number of applications of generalized Sylvester matrix equations in systems and control theory, in this paper we establish some necessary and sufficient conditions for the solvability to a system of eight generalized Sylvester matrix equations over the quaternion algebra. The general solution to this system is also constructed when it is solvable. Moreover, an algorithm and a numerical example are also given to make the results of this paper more practical in various fields of engineering. The findings of this paper generalize previous results in the literature. Mathematics Subject Classification. 15A03, 15A09, 15B33, 15A24, 15B57, 11R52. Keywords. Quaternion, Generalized Sylvester matrix equation, Exclusive solution, Moore–Penrose inverse, Rank.

1. Introduction Throughout this paper, we denote the quaternion algebra by H ={s0 + s1 i + s2 j + s3 k | i2 = j 2 = k 2 = ijk = −1, s0 , s1 , s2 , s3 ∈ R} where R is the real number field. The set of all m × n matrices over H is expressed by Hm×n . For a matrix A over H, A∗ represents the conjugate transpose of A. The Moore-Penrose inverse of A is denoted by A† if AA† A = A, A† AA† = A† , (AA† )∗ = AA† , (A† A)∗ = A† A. Furthermore, LA := I − A† A and RA := I − AA† are the projectors of A, respectively and LA = † (LA )∗ = (LA )2 = L†A , RA = (RA )2 = (RA )∗ = RA . Quaternions were first introduced by Sir William Rowan Hamilton, an Irish mathematician, in [10]. Clearly, H is associative and noncommutative. The quaternion matrices can be used in quantum physics, mechanics and signal processing [1,18,28]. A quaternion matrix is a sum of two complex ∗ Corresponding

author.

Author's personal copy A. Rehman et al.

Adv. Appl. Clifford Algebras

matrices which shows that if a result holds for a quaternion matrix then these findings must be true for complex matrix as well as for real matrix. In other words, real and complex number fields are subsets of quaternion algebra. Many problems in diverse fields of engineering like singular system control [9], system design [27] and linear descriptor system [6] require the solutions of generalized Sylvester matrix equations. The applications of generalized Sylvester matrix equations in perturbation theory, sensitivity analysis, neural networks and feedback can be viewed in [4,20,36] and [26]. For instance, Bai [2] examined the iterative solution of A1 X + XA2 = B. Roth researched the solvability condition of A1 X + Y A2 = B

(1.1)

in [25]. The general solution of (1.1) was established by Bakasalary and Kala in [3]. The researchers in [31] obtained the constraint solution of (1.1) with its general solution when some necessary and sufficient condition are fulfilled. Wang and He [32] obtained the necessary and sufficient conditions of A1 X1 + Z1 B1 = C1 , A2 X2 + Z1 B2 = C2 ,

(1.2)

to have a solution. They also derived the general solution of (1.2) when this system is consistent. Some solvability conditions to (1.2) were presented in [19]. The condition number of (1.2) was researched in [21]. The constraint solution of (1.2) was constructed by Wang et al. in [29]. Some solvability conditions of (1.2) with the condition X2 = X1 was established in [34]. The solution of (1.2) under the same condition was computed by a researcher in [14]. Wang and He also gave some necessary and sufficient conditions for A1 X1 + Z1 B1 = C1 , A2 Z1 + X2 B2 = C2 ,

(1.3)

to have a solution, and its general solution in [13]. The general solutions of some system of mixed type generalized Sylvester matrix equations involving four variable matrix are presented in [30]. Recently, the researchers in [11] obtained the general solution of A1 X1 + Z1 B1 = C1 , A2 X2 + Z1 B2 = C2 , A3 X2 + Z2 B3 = C3 , A4 X3 + Z2 B4 = C4 , by using rank equalities and generalized inverses when this system is consistent. The general solutions of some systems of matrix equations can be seen in [7,8,15–17,23,24]. The general solution of A1 X1 + Y1 B1 + C1 Z1 D1 = E1 , A2 X2 + Y2 B2 + C2 Z1 D2 = E2

(1.4)

was established in [35] when some solvability conditions are satisfied. Motivated by the intensive applications of generalized Sylvester matrix equations and recent development of quaternion matrices, in this paper we investigate some necessary and sufficient conditions of A3 X1 = C3 , Y1 B3 = C5 , F1 Z1 = G1 , Z1 F2 = G2 ,

Author's personal copy A Constraint System of Generalized Sylvester A4 X2 = C4 , Y2 B4 = C6 , A1 X1 + Y1 B1 + C1 Z1 D1 = E1 , A2 X2 + Y2 B2 + C2 Z1 D2 = E2

(1.5)

with its general solution when this system is solvable. Obviously, (1.2), (1.3) and (1.4) are particular cases of (1.5). The primary goal of this work is to establish some solvability conditions to (1.5) and an expression of its general solution by using generalized inverse and rank equalities of the given coefficient matrices. The remainder of this paper is organized as follows. In Sect. 2, we give some necessary and sufficient conditions for the consistence of (1.5) and an expression of the general solution to the system. Moreover, it is worth pointing out that the conditions in the main theorem can be easily verified by using our Maple package for quaternion computing. We give an algorithm with a numerical example to illustrate that our results are feasible in Sect. 3. Finally, a brief conclusion is given in Sect. 4.

2. Main Result We initiate with some known results. Lemma 2.1. [22]. Let K ∈ Hm×n , P ∈ Hm×t , Q ∈ Hl×n . Then 

   K rank − rank(QLK ) = rank(K), rank K P − rank(RP K) = rank(P ), Q   K P rank − rank(P ) − rank(Q) = rank(RP KLQ ). Q 0

Lemma 2.2. [5]. Let A1 and A2 be given matrices with adequate shapes. Then A1 X = A2 is solvable if and only if A2 = A1 A†1 A2 . Then its general solution is X = A†1 A2 + LA1 U1 , where U1 is a any matrix with conformable dimension. Lemma 2.3. [5]. Let B11 and D11 be known matrices with adequate dimen† sions. Then Y B11 = D11 is consistent if and only if D11 = D11 B11 B11 . Under this condition, its general solution is † + W1 RB11 , Y = D11 B11

where W1 is a free matrix with feasible dimension. Lemma 2.4. [33]. Let A1 ∈ Hm1 ×n1 , B1 ∈ Hr1 ×s1 , C1 ∈ Hm1 ×r1 , C2 ∈ Hn1 ×s1 be given and X1 ∈ Hn1 ×r1 is to be determined. Then the system A1 X1 = C1 , X1 B1 = C2 , is consistent if and only if RA1 C1 = 0, C2 LB1 = 0, A1 C2 = C1 B1 .

(2.1)

Author's personal copy A. Rehman et al.

Adv. Appl. Clifford Algebras

Under these conditions, a general solution to (2.1) can be established as X1 = A†1 C1 + LA1 C2 B1† + LA1 U1 RB1 , where U1 is a free matrix over H with suitable shape. Lemma 2.5. [12]. Let A1 , B1 , C3 , D3 , C4 , D4 and E1 be known. Assign A = RA1 C3 , B = D3 LB1 , C = RA1 C4 , D = D4 LB1 , E = RA1 E1 LB1 , F = RA C, G = DLB , H = CLF . Then the statements mentioned below are equivalent: (1) A1 U + V B1 + C3 W D3 + C4 ZD4 = E1 has a solution. (2) RF RA E = 0, ELB LG = 0, RA ELD = 0, RC ELB = 0. (3)   E1 C4 C3 A1 = rank(B1 ) + rank[C4 C3 A1 ], rank B1 0 0 0 ⎡ ⎤ ⎤ ⎡ E1 A1 D3 ⎢ D3 0 ⎥ ⎥ ⎦ ⎣ rank ⎢ ⎣ D4 0 ⎦ = rank D4 + rank(A1 ), B1 B1 0 ⎤ ⎡   E1 C3 A1 D4 0 ⎦ = rank[A1 C3 ] + rank rank ⎣ D4 0 , B1 0 B1 0 ⎤ ⎡   E1 C4 A1 D3 0 ⎦ = rank[A1 C4 ] + rank rank ⎣ D3 0 , B1 0 B1 0

(2.2)

(2.3)

(2.4)

(2.5)

(2.6)

In these conditions, a general solution to (2.2) is U = A†1 (E1 − C3 W D3 − C4 ZD4 ) − A†1 S7 B1 + LA1 S6 , V

= RA1 (E1 − C3 W D3 − C4 ZD4 )B1† + A1 A†1 S7 + S8 RB1 ,

W = A† EB † − A† CF † EB † − A† HC † EG† DB †

− A† HS2 RG DB † + LA S4 + S5 RB , Z = F † ED† + H † HC † EG† + LF LH S1 + LF S2 RG + S3 RD , where S1 , . . . , S8 are any matrices of viable shapes over H. Lemma 2.5 has a solid role in obtaining the main theorem of this paper. Theorem 2.1. Given A1 , A2 , A3 , A4 , B1 , B2 , B3 , B4 , C1 , C2 , C3 , C4 , C5 , C6 , D1 , D2 , E1 , E2 , F1 , F2 , G1 and G2 matrices of feasible shapes over H. Assign A5 = A1 LA3 , B5 = RB3 B1 , C7 = C1 LF1 , D7 = RF2 D1 , E3 = E1 − (A1 A†3 C3 + C5 B3† B1 + C1 F1† G1 D1 + C1 LF1 G2 F2† D2 ),

Author's personal copy A Constraint System of Generalized Sylvester A6 = RA5 C7 , B6 = D7 LB5 , E4 = RA5 E3 LB5 , A7 = A2 LA4 , E5 = E2 − (A2 A†4 C4 + C6 B4† B2 + C2 F1† G1 D2 + C2 LF1 G2 F2† D2 ), B7 = RB4 B2 , C8 = C2 LF1 , D8 = RF2 D2 , A = RA7 C9 , B = D8 LB7 , C = RA7 C8 , D = D9 LB7 , E = RA7 E5 LB7 , F = RA C, G = DLB , H = CLF .

(2.7)

Then the following statements are equivalent: (1) System (1.5) is consistent. (2) RA3 C3 = 0, RA4 C4 = 0, C5 LB3 = 0, C6 LB4 = 0, RF1 G1 = 0, G2 LF2 = 0, F1 G2 = G1 F2 , RA6 E4 = 0, E4 LB6 = 0, RF RA E = 0, ELB LG = 0, RA ELD = 0, RC ELB = 0.

(2.8)

(3)     rank A3 C3 = rank(A3 ), rank A4 C4 = rank(A4 ),     C5 C6 rank = rank(B3 ), rank = rank(B4 ), B3 B4     G2 rank F1 G1 = rank(F1 ), rank = rank(F2 ), F2 ⎤ ⎡ E1 C5 A1 C1 ⎢ 0 0 B1 B3 ⎥ ⎥ F1 G2 = G1 F2 , rank ⎢ ⎣ A3 0 C3 0 ⎦ 0 F1 G1 D1 0 ⎤ ⎡ A1 C1   = rank ⎣ A3 0 ⎦ + rank B1 B3 , 0 F1 ⎡ ⎤ E1 A1 C1 G1 C5     ⎢ D1 0 F2 0 ⎥ ⎥ = rank D1 F2 0 + rank A1 , rank ⎢ ⎣ B1 0 0 B3 ⎦ A3 B1 0 B3 C3 A3 0 0 ⎡ ⎤ ⎤ ⎡ E2 C2 A2 C6 C2 A2 ⎢ B2 ⎥   0 0 B4 ⎥ rank ⎢ = rank ⎣ F1 0 ⎦ + rank B2 B4 , ⎣ C2 G1 F1 0 ⎦ 0 0 A4 0 A4 0 C4 ⎡ ⎤ E2 A2 C2 G2 C6     ⎢ D2 0 F2 0 ⎥ ⎥ = rank D2 F2 0 + rank A2 , rank ⎢ ⎣ B2 0 0 B4 ⎦ A4 B2 0 B4 C4 A4 0 0

Author's personal copy A. Rehman et al.

Adv. Appl. Clifford Algebras

⎡

⎤ E2 C2 A2 0 C6 0 0 0 ⎢ D2 0 0 D1 0 0 F2 0 ⎥ ⎢ ⎥ ⎢ B2 ⎥ 0 0 0 B 0 0 0 4 ⎢ ⎥ ⎢ 0 ⎥ 0 E 0 A C G 0 C 1 1 1 1 2 ⎢ ⎥ rank ⎢ ⎥ C 0 A 0 0 0 0 0 4 4 ⎢ ⎥ ⎢ 0 ⎥ 0 0 0 B 0 0 B 1 3⎥ ⎢ ⎣ G1 D2 F1 0 0 0 0 0 0 ⎦ 0 0 0 0 0 0 0 A3 ⎡ ⎤ A2 C2 0 ⎡ ⎢ A4 0 D2 D1 0 F2 0 ⎥ ⎢ ⎥ ⎥ ⎣ A 0 C B2 0 B4 0 = rank ⎢ + rank 1 1⎥ ⎢ ⎣ 0 F1 0 ⎦ 0 0 B1 0 0 0 A3 ⎡ ⎤ E2 C2 A2 C2 G2 C6 ⎢ D2 0 0 F2 0 ⎥ ⎢ ⎥ ⎥ 0 0 0 B B rank ⎢ 4⎥ ⎢ 2 ⎣ 0 F1 0 0 0 ⎦ 0 0 C4 0 A4 ⎡ ⎤   C2 A2 D2 F2 0 = rank ⎣ F1 0 ⎦ + rank . B2 0 B4 0 A4

⎤ 0 0 ⎦, B3

(2.9)

In this case, the general solution to (1.5) can be expressed as X1 = A†3 C3 + LA3 U1 ,

(2.10)

Y1 =

(2.11)

Z1 = U1 = V1 = X2 = Y2 = W = U2 = V2 =

C5 B3† + V1 RB3 , F1† G1 + LF1 G2 F2† + LF1 W RF2 , A†5 (E3 − C7 W D7 ) − A†5 T4 B5 + LA5 T5 , RA5 (E3 − C7 W D7 )B5† + A5 A†5 T4 + T6 RB5 , A†4 C4 + LA4 U2 , C6 B4† + V2 RB4 , A†6 E4 B6† + LA6 W1 + W2 RB6 , A†7 (E4 − C9 W1 D8 − C8 W2 D9 ) − A†7 T1 B7 + LA7 T2 , RA7 (E4 − C9 W1 D8 − C8 W2 D9 )B7† + A7 A†7 T1 + T3 RB7 ,

(2.12)

(2.13) (2.14) (2.15) (2.16)

with W1 = A† EB † − A† CF † EB † − A† HC † EG† DB † −A† HZ2 RG DB † + LA Z3 + Z4 RB , †

†

†

†

(2.17)

†

W2 = F ED + H HC EG + LF LH Z5 + LF Z2 RG + Z6 RD , (2.18) where T1 , . . . , T6 , Z2 , . . . , Z6 are any matrices of comformable sizes over H. Proof. (1) ⇐⇒ (2): We write the equations (1.5) as follows: A3 X1 = C3 , Y1 B3 = C5 ,

Author's personal copy A Constraint System of Generalized Sylvester F1 Z1 = G1 , Z1 F2 = G2 , A1 X1 + Y1 B1 + C1 Z1 D1 = E1 ,

(2.19)

A4 X2 = C4 , Y2 B4 = C6 , A2 X2 + Y2 B2 + C2 Z1 D2 = E2 .

(2.20)

and

By Lemmas 2.2, 2.3 and 2.4, the general solutions to A3 X1 = C3 , Y1 B3 = C5 and F1 Z1 = G1 , Z1 F2 = G2 are X1 = A†3 C3 + LA3 U1 ,

(2.21)

Y1 = C5 B3† + V1 RB3 ,

(2.22)

Z1 = F1† G1 + LF1 G2 F2† + LF1 W RF2

(2.23)

and respectively, where U1 , V1 and W are arbitry matrices of suitable sizes. By using (2.21)–(2.23) to the last equation of (2.19), we obtain (2.24)

A5 U1 + V1 B5 + C7 W D7 = E3 .

By Lemma 2.5, Eq. (2.24) has a solution if and only if RA6 E4 = 0, E4 LB6 = 0. The general solutions to Eq. (2.24) is U1 = A†5 (E3 − C7 W D7 ) − A†5 T4 B5 + LA5 T5 ,

V1 = RA5 (E3 − C7 W D7 )B5† + A5 A†5 T4 + T6 RB5 ,

and W = A†6 E4 B6† + LA6 W1 + W2 RB6

(2.25)

respectively, where T4 , T5 , T6 , W1 and W2 are arbitrary matrices of compatible sizes over H. Similarly, the general solutions to A4 X2 = C4 and Y2 B4 = C6 by Lemmas 2.2 and 2.3 are X2 = A†4 C4 + LA4 U2

(2.26)

Y2 = C6 B4† + V2 RB4

(2.27)

and respectively, where U2 and V2 are any matrices with feasible shapes. By using (2.25)–(2.27) to the last equation of (2.20), we obtain A7 U2 + V2 B7 + C9 W1 D8 + C8 W2 D9 = E5 .

(2.28)

By Lemma 2.5, Eq. (2.28) has a solution if and only if the last four equations of (2.8) are satisfied and in this case its general solution can be expressed by (2.15)–(2.18), respectively. (2) ⇐⇒ (3) Now we prove that all the conditions mentioned in (2.8) are equiavelnt to all the rank equalities mentioned in (2.9). We just show that the last equality in (2.8) is equivalent to the last rank equality described in (2.9) and the other equalities can be shown in the same way. Now applying Lemma 2.1 and the elementary row operation to rank(RC ELB ), we have

Author's personal copy A. Rehman et al.

Adv. Appl. Clifford Algebras



 E C = rank − rank(B) − rank(C) B 0   RA7 E5 LB7 RA7 C8 = rank − rank(RA7 C8 ) − rank(D8 LB7 ) D8 LB7 0 ⎤ ⎡   E5 C8 A7   D8 0 ⎦ − rank C8 A7 − rank = rank ⎣ D8 0 B7 0 B7 0 ⎤ ⎡ E5 C2 LF1 A2 LA4 0 0 ⎦ = rank ⎣ RF2 D2 0 0 RB4 B2     RF2 D2 −rank C2 LF1 A2 LA4 − rank RB4 B2 ⎡ ⎤ E5 C2 A2 0 0 ⎢ D2 0 0 F 0 ⎥ 2 ⎢ ⎥ ⎥ 0 0 0 B B = rank ⎢ 4⎥ ⎢ 2 ⎣ 0 F1 0 0 0 ⎦ 0 0 0 A4 0 ⎡ ⎤   C2 A2 D2 F2 0 −rank ⎣ F1 0 ⎦ − rank B2 0 B4 0 A4 ⎡ ⎤ E2 C2 A2 C2 G2 C6 ⎢ D2 0 0 F2 0 ⎥ ⎢ ⎥ ⎢ 0 0 B4 ⎥ = rank ⎢ B2 0 ⎥ ⎣ 0 F1 0 0 0 ⎦ 0 0 C4 0 A4 ⎡ ⎤   C2 A2 D2 F2 0 −rank ⎣ F1 0 ⎦ − rank B2 0 B4 0 A4 Hence rank(RC ELB ) = 0 is the same as the last rank equality of (2.9).



Now we consider some special cases of the system (1.5). If A3 , C3 , B3 , C5 , F1 , G1 , F2 , G2 , A4 , C4 , B4 and C6 all have value equal to zero in Theorem 2.1, then we get the following outcome. Corollary 2.1. Given A1 , A2 , B1 , B2 , C1 , C2 , D1 , D2 , E1 and E2 matrices of feasible shapes over H. Assign A3 = RA1 C1 , B3 = D1 LB1 , C3 = RA1 E1 LB1 , B4 = RB3 D2 , C4 = E2 − C2 A†3 C3 B3† D2 , A = RA2 A4 , B = D2 LB2 , C = RA2 C2 , D = B4 LB2 , E = RA2 C4 LB2 , F = RA C, G = DLB , H = CLF . Then system (1.4) is consistent if and only if (2) RA3 C3 = 0, C3 LB3 = 0, RF RA E = 0, ELB LG = 0, RA ELD = 0, RC ELB = 0.

Author's personal copy A Constraint System of Generalized Sylvester Under these condition, the general solution to (1.4) is X1 = A†1 (E1 − C1 Z1 D1 ) − A†1 T1 B1 + LA1 T2 ,

Y1 = RA1 (E1 − C1 Z1 D1 ) + A1 A†1 T1 + T3 RB1 ,

Z1 = A†3 C3 B3† + LA3 W1 + W2 RB3 ,

X2 = A†2 (C4 − A4 W1 D2 − C2 W2 B4 ) − A†2 T4 B2 + LA2 T5 ,

Y2 = RA2 (C4 − A4 W1 D2 − C2 W2 B4 )B2† + A2 A†2 T4 + T6 RB2 ,

with W1 = A† EB † − A† CF † EB † − A† HC † EG† DB † − A† HZ2 RG DB † + LA Z3 + Z4 RB , W2 = F † ED† + H † HC † EG† + LF LH Z5 + LF Z2 RG + Z6 RD , where T1 , T2 , T3 , T4 , T5 , T6 , Z2 , . . . , Z6 are any matrices of suitable shapes over H. Remark 2.1. The above Corollary has the main findings of [35]. If A3 , C3 , B3 , C5 , F1 , G1 , F2 , G2 , A4 , C4 , B4 , C6 , B1 , B2 all have value equal to zero and Ci = I, (i = 1, 2) in Theorem 2.1, then we arrive at the following finding. Corollary 2.2. Suppose that Ai , Bi and Ci , (i = 1, 2) are given. Put A = RB1 , B = RA2 A1 , C = B2 LA , D = RA2 (C2 − RA1 C1 B1† B2 )LA . Then (1.2) has a solution if and only if RA1 C1 LB1 = 0, RB D = 0, DLC = 0. Under these conditions, the general solution to (1.2) is X1 = A†1 C1 − W1 B1 + LA1 W2 ,

Z1 = RA1 C1 B1† + A1 W1 + W3 RB1 ,

X2 = A†2 (C2 − RA1 C1 B1† B2 − A1 W1 B2 ) − W4 A + LA2 W5 with W1 = B † DC † + LB W6 + W7 RC , W3 = RA2 (C2 − RA1 C1 B1† B2 − A1 W1 B2 )A† + A2 W4 + W8 RA , where W2 , W4 , . . . , W8 are arbirtary matrices over H of conformable shapes. Remark 2.2. The above Corollary has the main findings of [32]. If A3 , C3 , B3 , C5 , F1 , G1 , F2 , G2 , A4 , C4 , B4 , C6 , B1 , B2 are all have value equal to zero and C1 = I, D2 = I in Theorem 2.1, then we get the general solution to (1.3), which was researched in [13].

Author's personal copy A. Rehman et al.

Adv. Appl. Clifford Algebras

Corollary 2.3. Suppose that Ai , Bi and Ci , (i = 1, 2) are given. Put A = RA2 A1 , B = RB1 LB2 , C = RA2 A1 (C2 − A2 RA1 C1 B1† )LB2 . Then (1.3) is solvable if and only if RA1 C1 LB1 = 0, RA C = 0, CLB = 0. Under these circumstances, the general solution to (1.3) is X1 = A†1 C1 − W1 B1 + LA1 W2 ,

Z1 = RA1 C1 B1† + A1 W1 + W3 RB1 ,

X2 = RA2 A1 (C2 − A2 RA1 C1 B1† − A2 W3 RB1 )B2† + A2 A1 W4 + W5 RB2 with W3 = A† CB † + LA W6 + W7 RB , W1 = (A2 A1 )† (C2 − A2 RA1 C1 B1† B2 − A2 W3 RB1 ) − W4 B2 + LA2 A1 W8 , where W2 , W4 , . . . , W8 are free matrices of suitable shapes over H.

3. Algorithm with a Numerical Example Using mathematics software Maple, we have developed a Maple package for quaternion computations. The rank equalities in above theorem can be quickly calculated and verified. Here is an example to illustrate the Theorem 2.1. Algorithm 3.1: (1) Feed the values of Ai , Bj , (i, j = 1, . . . , 4), C1 , C2 , D1 , D2 , E1 , E2 , F1 , F2 , G1 and G2 with conformable shapes over H. (2) Compute A5 , B5 , A6 , B6 , C7 , D7 , A7 , B7 , C8 , D8 , C9 , D9 , A, B, C, D, E, F, G and H by (2.7). (3) Check (2.8) or (2.9) is ok or not. If no then return “inconsistent”. (4) Else, compute X1 , Y1 , Z1 , X2 and Y2 by (2.10)–(2.14). Example 3.2 For given matrices ⎡ ⎤    j k 1 2j i 2 ⎣ ⎦ A1 = , B1 = 1 2 , C1 = j k 2 1 i j ⎡ ⎤ ⎡   i j 1 i 2 3 A2 = ⎣ k 0 ⎦ , B2 = , C2 = ⎣ i 1 j k 1 i k ⎡     1 1 i 0 1 2 A3 = , C3 = , B3 = ⎣ j 2 0 j j k 0 ⎡     1 i 1 2 1 i F1 = , G1 = , A4 = ⎣ i 2 k j j 2 j

   j 1 j , D1 = , k i k ⎤   j 2 1 j k ⎦ 0 3 , D2 = , i 0 2 i j ⎤   i 1 j ⎦ 2 , C5 = , i 0 1 ⎤ ⎡ ⎤ k 1 i j 0 ⎦ , C4 = ⎣ k 0 1 ⎦ , 2 0 j 2 i 0

Author's personal copy A Constraint System of Generalized Sylvester  B4 =

i k



⎡

1 j , C6 = ⎣ j 1 k

⎤  2 i 0 ⎦ , F2 = 2 i

⎤ 1 k k , G2 = ⎣ j 2 ⎦ . j 0 i 

⎡

Upon calculation, we have that (2.9) holds and B6 = 0, A7 = 0, D8 = 0, C9 = 0, D9 = 0, A = 0, B = 0, D = 0, F = 0, G = 0, H = 0. Hence the general solution to (1.5) can be expressed as ⎡ ⎤ i 2 X 1 = ⎣ i + j 3 ⎦ + M 1 T 4 M2 , 1 k   k 1 i Y1 = + N1 T 4 N2 , 2 j 1+k ⎡ ⎤ ⎡ ⎤ i 1 i 1 0 ⎦ , X2 = ⎣ j 2 ⎦, Z1 = ⎣ k + j 0 i+j k i+j     i j i 3 Y2 = + T3 , k 2+i 4 k where T4 and T3 are free matrices of adequate sizes and ⎡

⎤ 0.0270 −0.1622j M1 = ⎣ 0.0270i −0.1622k ⎦ , 0.0541j 0.3243 ⎡ ⎤ 0.1429 + 0.2857j + 0.2857k 0.2857 + 0.0000i + 0.2857j + 0.0000k 0.2857 + 0.0000i − 0.2857j + 0.0000k ⎦ , M2 = ⎣ 0.2857 − 0.2857i − 0.1429j −0.5714 + 0.7143i + 0.2857k −0.5714 + 0.2857i + 0.5714j + 0.2857k   0.0270 −0.1622j N1 = , 0.1622j 0.9730 ⎡ ⎤ 0.1429 + 0.0000i 0.1429j + 0.0000k −0.1429i − 0.2857j −0.2857 − 0.1429k ⎦ . N2 = ⎣ 0.0000i − 0.1429j 0.1429 + 0.0000k 0.1429i + 0.2857j −0.2857 + 0.1429k 0.7143 + 0.0000k

4. Conclusion We establish some necessary and sufficient conditions for the solvability to the system (1.5) over H. The expression of the general solution to (1.5) is also constructed when it is solvable. In summary, it concludes that the research presented in this paper extends the previous results of [13,32,35]. An algorithm and a numerical example are also given to illustrate the results of this paper. Acknowledgements This research is supported by the Grant from the National Natural Science Foundation of China (11571220).

Author's personal copy A. Rehman et al.

Adv. Appl. Clifford Algebras

References [1] Adler, S.L.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, New York (1995) [2] Bai, Z.Z.: On Hermitian and skew-Hermitian splitting iteration methods for continuous Sylvester equations. J. Comput. Math. 29, 185–198 (2011) [3] Baksalary, J.K., Kala, R.: The matrix equation AX − Y B = C. Linear Algebra Appl. 25, 41–43 (1979) [4] Barraud, A., Lesecq, S., Christov, N.: From sensitivity analysis to random floating point arithmetics-application to Sylvester equations. Numer. Anal. Appl. Lect. Notes Comput. Sci. 2001, 35–41 (1988) [5] Buxton, J.N., Churchouse, R.F., Tayler, A.B.: Matrices Methods and Applications. Clarendon Press, Oxford (1990) [6] Darouach, M.: Solution to Sylvester equation associated to linear descriptor systems. Syst. Control Lett. 55, 835–838 (2006) [7] Dmytryshyn, A., K˚ astr¨ om, B.: Coupled Sylvester-type matrix equations and block diagonalization. SIAM J. Matrix Anal. Appl. 38, 580–593 (2015) [8] Farid, F.O., He, Z.H., Wang, Q.W.: The consistency and the exact solutions to a system of matrix equations. Linear Multilinear Algebra 64(11), 2133–2158 (2016) [9] Gavin, K.R., Bhattacharyya, S.P.: Robust and well-conditioned eigenstructure assignment via Sylvester’s equation. In: Proceedings of American Control Conference (1982) [10] Hamilton, W.R.: On quaternions; or on a new system of imaginaries in algebra. Philos. Mag. 25(3), 489–495 (1844) [11] He, Z.H., Wang, Q.W.: A System of periodic discrete-time coupled Sylvester quaternion matrix equations. Algebra Coll. 24(1), 169–180 (2017) [12] He, Z.H., Wang, Q.W.: The general solutions to some systems of matrix equations. Linear Multilinear Algebra 63(10), 2017–2032 (2015) [13] He, Z.H., Wang, Q.W.: A pair of mixed generalized Sylvester matrix equations. J. Shanghai Univ. Nat. Sci. 20, 138–156 (2014) [14] K˚ agstr¨ om, B.: A perturbation analysis of the generalized Sylvester equation (AR − LB, DR − LE) = (C, F ). SIAM J. Matrix Anal. Appl. 15, 1045–1060 (1994) [15] Kyrchei, I.I.: Determinantal representation of the Moore–Penrose inverse matrix over the quaternion skew field. J. Math. Sci. 180(1), 23–33 (2012) [16] Kyrchei, I.I.: Explicit representation formulas for the minimum norm least squares solutions of some quaternion matrix equations. Linear Algebra Appl. 438(1), 136–152 (2013) [17] Kyrchei, I.I.: Weighted singular value decomposition and determinantal representations of the quaternion weighted Moore-Penrose inverse. Appl. Math. Comput. 309, 1–16 (2017) [18] Leo, S.D., Scolarici, G.: Right eigenvalue equation in quaternionic quantum mechanics. J. Phys. A. 33, 2971–2995 (2000) [19] Lee, S.G., Vu, Q.P.: Simultaneous solutions of matrix equations and simultaneous equivalence of matrices. Linear Algebra Appl. 437, 2325–2339 (2012)

Author's personal copy A Constraint System of Generalized Sylvester [20] Li, R.C.: A bound on the solution to a structured Sylvester equation with an application to relative perturbation theory. SIAM J. Matrix Anal. Appl. 21(2), 440–445 (1999) [21] Lin, Y.Q., Wei, Y.M.: Condition numbers of the generalized Sylvester equation. IEEE Trans. Autom. Control 52, 2380–2385 (2007) [22] Marsaglia, G., Styan, G.P.H.: Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra 2, 269–292 (1974) [23] Rehman, A., Wang, Q.W.: A system of matrix equations with five variables. Appl. Math. Comput. 271, 805–819 (2015) [24] Rehman, A., Wang, Q.W., He, Z.H.: Solution to a system of real quaternion matrix equations encompassing η-Hermicity. Appl. Math. Comput. 265, 945– 957 (2015) [25] Roth, W.E.: The equations AX − Y B = C and AX − XB = C in matrices. Proc. Am. Math. Soc. 3, 392–396 (1952) [26] Syrmos, V.L., Lewis, F.L.: Output feedback eigenstructure assignment using two Sylvester equations. IEEE Trans. Autom. Control 38, 495–499 (1993) [27] Syrmos, V.L., Lewis, F.L.: Coupled and constrained Sylvester equations in system design. Circ. Syst. Signal Process. 13(6), 663–694 (1994) [28] Took, C.C., Mandic, D.P.: Augmented second-order statistics of quaternion random signals. Signal Process. 91, 214–224 (2011) [29] Wang, Q.W., Rehman, A., He, Z.H., Zhang, Y.: Constraint generalized Sylvester matrix equations. Automatica 69, 60–64 (2016) [30] Wang, Q.W., He, Z.H.: Systems of coupled generalized Sylvester matrix equations. Automatica 50, 2840–2844 (2014) [31] Wang, L., Wang, Q.W., He, Z.H.: The common solution of some matrix equations. Algebra Coll. 23(1), 71–81 (2016) [32] Wang, Q.W., He, Z.H.: Solvability conditions and general solution for the mixed Sylvester equations. Automatica 49, 2713–2719 (2013) [33] Wang, Q.W., Wu, Z.C., Lin, C.Y.: Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications. Appl. Math. Comput. 182, 1755–1764 (2006) [34] Wimmer, H.K.: Consistency of a pair of generalized Sylvester equations. IEEE Trans. Autom. Control 39, 1014–1016 (1994) [35] Zhang, X.: A system of generalized Sylvester quaternion matrix equations and its applications. Appl. Math. Comput. 273, 74–81 (2016) [36] Zhang, Y.N., Jiang, D.C., Wang, J.: A recurrent neural network for solving Sylvester equation with time-varying coefficients. IEEE Trans. Neural Netw. 13(5), 1053–1063 (2002)

Abdur Rehman, Ilyas Ali and Muhammad Akram University of Engineering and Technology Lahore Pakistan e-mail: abdur rehman [email protected]

Author's personal copy A. Rehman et al. Qing-Wen Wang Department of Mathematics Shanghai University Shanghai 200444 People’s Republic of China e-mail: [email protected]; [email protected] M. O. Ahmad The University Of Lahore Lahore Pakistan Received: March 24, 2017. Accepted: August 2, 2017.

Adv. Appl. Clifford Algebras