A New Solution to the Matrix Equation 𝑋−𝐴𝑋𝐵=𝐶

Abstract

We investigate the matrix equation 𝑋−𝐴𝑋𝐵=𝐶. For convenience, the matrix equation 𝑋−𝐴𝑋𝐵=𝐶 is named as Kalman-Yakubovich-conjugate matrix equation. The explicit solution is constructed when the above matrix equation has unique solution. And this solution is stated as a polynomial of coefficient matrices of the matrix equation. Moreover, the explicit solution is also expressed by the symmetric operator matrix, controllability matrix, and observability matrix. The proposed approach does not require the coefficient matrices to be in arbitrary canonical form. At the end of this paper, the numerical example is shown to illustrate the effectiveness of the proposed method.

1. Introduction

Throughout this paper, we use 𝑅 and 𝐶 to denote the real number field and the complex number field. We use 𝐴^𝑇, 𝐴, 𝐴^𝐻, and 𝐴^∗ to denote transpose, conjugate, conjugate transpose, and the adjoint matrix of 𝐴, respectively. 𝜎(𝐴) and 𝜆(𝐵) are the sets of characteristic eigenvalues of matrices 𝐴 and 𝐵, respectively. 𝐼 is represented as appropriate dimension identity matrix. Moreover, for 𝐴∈𝐶^𝑛×𝑛, 𝐵∈𝐶^𝑛×𝑟, and 𝐶∈𝐶^𝑚×𝑛, we have the following notations: 𝑄_𝑐(𝐴,𝐵,𝑛)=𝐵𝐴𝐵⋯𝐴^𝑛−1𝐵,𝑄_𝑜()=𝐶⋮𝐴,𝐶,𝑘𝐶𝐴𝐶𝐴^𝑘−1,𝑓_𝐴(𝑠)=det(𝐼−𝑠𝐴)=𝛼_𝑛𝑠^𝑛+𝛼_𝑛−1𝑠^𝑛−1+⋯+𝛼₁𝑆𝑠+1,_𝑟()=𝐼𝐼,𝐴^𝑟𝛼₁𝐼_𝑟𝛼₂𝐼_𝑟⋯𝛼_𝑛−1𝐼_𝑟𝐼_𝑟𝛼₁𝐼_𝑟⋯𝛼_𝑛−2𝐼_𝑟𝐼⋯⋯_𝑟𝛼₁𝐼_𝑟𝐼_𝑟. (1) In this case, 𝑄_𝑐(𝐴,𝐵,𝑛), (𝐴,𝐵), 𝑄_𝑜(𝐴,𝐶,𝑘), and 𝑆_𝑟(𝐼,𝐴) are named as the controllability matrix, the observability matrix, and a symmetric operator matrix, respectively.

Matrix equations are often encountered in system theory and control theory, such as Lyapunov matrix equation, Sylvester matrix equation, and so on. The traditional method is that we convert this kind of matrix equations into their equivalent forms by using the Kronecker product, however, which involves the inversion of the associated large matrix and results in increasing computation and excessive computer memory. In the field of matrix algebra, some complex matrix equations have attached much attention from many researchers since it is shown in [1]. In [2, 3], the consistence of the matrix equation 𝐴𝑋−𝑋𝐵=𝐶 is related to the consimilarity of two partitioned matrices associated with the matrices 𝐴, 𝐵, and 𝐶. In the preceding matrix equation, 𝑋 denotes the matrix obtained by taking the complex conjugate of each element of 𝑋. Recently, in [4] some explicit expressions of the solution to the matrix equation 𝐴𝑋−𝑋𝐵=𝐶 were established by means of real representation of a complex matrix, and it is shown that there exists a unique solution if and only if 𝐴𝐴 and 𝐵𝐵 have no common eigenvalues. Yuan and Liao [5] investigated the least squares solution of the quaternion 𝑗-conjugate matrix equation 𝑋−𝐴𝑋𝐵=𝐶 (where 𝑋 denotes the 𝑗-conjugate of quaternion matrix 𝑋) with the least norm using the complex representation of a quaternion matrix, the Kronecker product of the matrices and the Moore-Penrose generalized inverse. The authors in [6] considered the matrix nearness problem associated with the quaternion matrix equation 𝐴𝑋𝐴^𝐻+𝐵𝑌𝐵^𝐻=𝐶  by means of the CCD-Q, GSVD-Q, and the projection theorem in the finite dimensional inner product space. In [7, 8], the solutions to matrix equations 𝑋−𝐴𝑋𝐵=𝐶 and 𝑋−𝐴𝑋𝐵=𝐶 have been expressed in terms of the characteristic polynomial of the matrix 𝐴. Song and Chen [9, 10] established the explicit solutions of the quaternion matrix equations 𝑋𝐹−𝐴𝑋=𝐶 and 𝑋−𝐴𝑋𝐹=𝐶, where 𝑋 denote the 𝑗-conjugate of the quaternion matrix. Wang et al. in [11–13] investigated the extreme ranks of the solutions to the quaternion matrix equations 𝐴𝑋𝐵=𝐶 and 𝐴𝑋𝐵+𝐶𝑌𝐷=𝐸, respectively, the equivalence canonical form of a matrix triplet over an arbitrary division ring, solvable condition for a pair of generalized Sylvester equations, and so on. Moreover, other matrix equations such as the coupled Sylvester matrix equations and the Riccati equations have also found numerous applications in control theory. For more related introduction, see [14–19] and the references therein.

In the present paper, we investigate the polynomial solutions to the Karm-Yakubovich-conjugate matrix equation. By the Faddeev Leverrier algorithm and characteristic polynomial, we provide the explicit solutions to the well-known Karm-Yakubovich-conjugate matrix equation. In addition, the equivalent forms of the solution have been proposed.

The rest of this paper is outlined as follows. In Section 2, the polynomial solutions to the Karm-Yakubovich-conjugate matrix equation 𝑋−𝐴𝑋𝐵=𝐶 are given. One of the polynomial solutions has a neat and elegant form in terms of symmetric operator matrix, controllability matrix, and observability matrix. The polynomial solution to the Karm-Yakubovich-conjugate is also proposed by the generalized Faddeev Leverrier algorithm. The example is given to show the efficiency of the proposed algorithm in Section 3.

2. Kalman-Yakubovich-Conjugate Matrix Equation 𝑋−𝐴𝑋𝐵=𝐶

In this section, the following matrix equation is investigated: 𝑋−𝐴𝑋𝐵=𝐶, (2) where 𝐴∈𝐶^𝑛×𝑛, 𝐵∈𝐶^𝑝×𝑝, and 𝐶∈𝐶^𝑛×𝑝.

Lemma 1. Assume that 𝐴∈𝐶^𝑛×𝑛, 𝐵∈𝐶^𝑝×𝑝, and 𝐶∈𝐶^𝑛×𝑝, if 𝑋 is a solution of (2). Then for any integer 𝑛−1≥𝑘≥0, the following conclusion can be established: 𝑋𝐵𝐵^𝑘−𝐴𝐴^𝑛−𝑘𝑋𝐵𝐵^𝑛=^𝑛−1_𝑗=𝑘𝐴𝐴^𝑗−𝑘𝐶𝐵𝐵^𝑗+^𝑛−1_𝑗=𝑘𝐴𝐴^𝑗−𝑘𝐴𝐶𝐵𝐵𝐵^𝑗. (3)

Proof. We prove this conclusion by mathematical induction. By postmultiplying both sides of (2) by (𝐵𝐵)^𝑛−1, we have 𝑋𝐵𝐵^𝑛−1−𝐴𝑋𝐵𝐵𝐵^𝑛−1=𝐶𝐵𝐵^𝑛−1. (4) By taking conjugate in both sides of (2), we can get 𝑋−𝐴𝑋𝐵=𝐶. (5) By postmultiplying both sides of (5) by 𝐵(𝐵𝐵)^𝑛−1 and premultiplying both sides of (5) by 𝐴, we have 𝐴𝑋𝐵𝐵𝐵^𝑛−1−𝐴𝐴𝑋𝐵𝐵𝐵𝐵^𝑛−1=𝐴𝐶𝐵𝐵𝐵^𝑛−1. (6) Combining (4) with (6), we can obtain 𝑋𝐵𝐵^𝑛−1−𝐴𝐴𝑋𝐵𝐵𝐵𝐵^𝑛−1=𝐶𝐵𝐵^𝑛−1+𝐴𝐶𝐵𝐵𝐵^𝑛−1. (7) This implies that relation (3) holds for 𝑘=𝑛−1. Now we assume that relation (3) holds for 𝑘=𝑁, (𝑛−1≥𝑁≥1); that is, 𝑋𝐵𝐵^𝑁−𝐴𝐴^𝑛−𝑁𝑋𝐵𝐵^𝑛=^𝑛−1_𝑗=𝑁𝐴𝐴^𝑗−𝑁𝐶𝐵𝐵^𝑗+^𝑛−1_𝑗=𝑁𝐴𝐴^𝑗−𝑁𝐴𝐶𝐵𝐵𝐵^𝑗. (8) Premultiplying both sides of (8) by 𝐴𝐴, we have 𝐴𝐴𝑋𝐵𝐵^𝑁−𝐴𝐴^{𝑛−𝑁+1}𝑋𝐵𝐵^𝑛=^𝑛−1_𝑗=𝑁𝐴𝐴^{𝑗−𝑁+1}𝐶𝐵𝐵^𝑗+^𝑛−1_𝑗=𝑁𝐴𝐴^{𝑗−𝑁+1}𝐴𝐶𝐵𝐵𝐵^𝑗. (9) By (5), we have 𝐴𝑋𝐵−𝐴𝐴𝑋𝐵𝐵=𝐴𝐶𝐵. (10) Combining (10) with (2), we can obtain 𝑋−𝐴𝐴𝑋𝐵𝐵=𝐶+𝐴𝐶𝐵. (11) According to (11), it is easy to obtain 𝑋𝐵𝐵^𝑁−1−𝐴𝐴𝑋𝐵𝐵^𝑁=𝐶+𝐴𝐶𝐵𝐵𝐵^𝑁−1. (12) Combining (9) with (12), we can obtain 𝑋𝐵𝐵^𝑁−1−𝐴𝐴^{𝑛−𝑁+1}𝑋𝐵𝐵^𝑛=^𝑛−1_{𝑗=𝑁−1}𝐴𝐴^{𝑗−(𝑁−1)}𝐶𝐵𝐵^𝑗+^𝑛−1_{𝑗=𝑁−1}𝐴𝐴^{𝑗−(𝑁−1)}𝐴𝐶𝐵𝐵𝐵^𝑗. (13) This implies that relation (3) holds for 𝑘=𝑁−1. So the conclusion is true.

Define _𝑘(𝐴,𝐶,𝐵)=^𝑛−1_𝑗=𝑘𝐴𝐴^𝑗−𝑘𝐶𝐵𝐵^𝑗+^𝑛−1_𝑗=𝑘𝐴𝐴^𝑗−𝑘𝐴𝐶𝐵𝐵𝐵^𝑗, (14) and then equality (3) can be compactly written as 𝑋𝐵𝐵^𝑘−𝐴𝐴^𝑛−𝑘𝑋𝐵𝐵^𝑛=_𝑘(𝐴,𝐶,𝐵). (15)

Let 𝑓_𝐴𝐴(𝑠)=det𝑠𝐼−𝐴𝐴=𝛼_𝑛+𝛼_𝑛−1𝑠+⋯+𝛼₀𝑠^𝑛, (16) and then 𝑓_(𝐴𝐴)(𝑠)=det𝐼−𝑠𝐴𝐴=𝛼₀+𝛼₁𝑠+⋯+𝛼_𝑛𝑠^𝑛. (17) It follows from (15), (16), and (17) that we have ^𝑛−1_𝑘=0𝛼_𝑘𝑋𝐵𝐵^𝑘−𝐴𝐴^𝑛−𝑘𝑋𝐵𝐵^𝑛=^𝑛−1_𝑘=0𝛼_𝑘𝑋𝐵𝐵^𝑘−𝛼_𝑘𝐴𝐴^𝑛−𝑘𝑋𝐵𝐵^𝑛+𝛼_𝑛𝑋𝐵𝐵^𝑛−𝑋𝐵𝐵^𝑛=^𝑛_𝑘=0𝛼_𝑘𝑋𝐵𝐵^𝑘−^𝑛_𝑘=0𝛼_𝑘𝐴𝐴^𝑛−𝑘𝑋𝐵𝐵^𝑛=𝑋𝑓_(𝐴𝐴)𝐵𝐵−𝑓_𝐴𝐴𝐴𝐴𝑋𝐵𝐵^𝑛=𝑋𝑓_(𝐴𝐴)𝐵𝐵. (18) On the other hand, ^𝑛−1_𝑘=0𝛼_𝑘𝑋𝐵𝐵^𝑘−𝐴𝐴^𝑛−𝑘𝑋𝐵𝐵^𝑛=^𝑛−1_𝑘=0𝛼_𝑘𝑘(𝐴,𝐶,𝐵). (19) Now, we define (𝐴,𝐶,𝐵)=^𝑛−1_𝑘=0𝛼_𝑘𝑘(𝐴,𝐶,𝐵). (20) It is obvious that ∏(𝐴,𝐶,𝐵) is a polynomial of the matrices 𝐴, 𝐵, and 𝐶. This polynomial is defined by the coefficient matrices and the characteristic polynomial of 𝐴𝐴. That is to say, for each Kalman-Yakubovich-conjugate equation of the form (2), there is a uniquely determined polynomial ∏(𝐴,𝐶,𝐵) of its coefficients matrices. Therefore, we obtain the following equation: 𝑋𝑓_𝐴𝐴𝐵𝐵=(𝐴,𝐶,𝐵). (21)

Theorem 2. If 𝜂𝛾≠1, for any 𝜂∈𝜆(𝐴𝐴) and 𝛾∈𝜆(𝐵𝐵), then (21) is equivalent to (2).

Proof. From the abovementioned argument, it is shown that (2) implies (21). Now we prove that (21) implies (2) when 𝜂𝛾≠1 for any 𝜂∈𝜆(𝐴𝐴) and 𝛾∈𝜆(𝐵𝐵). Now suppose that 𝑋 is a solution of (21), we can obtain 𝑋𝑓_𝐴𝐴𝐵𝐵−𝐴𝐴𝑋𝐵𝐵𝑓_𝐴𝐴=𝐵𝐵𝑋−𝐴𝐴𝑋𝐵𝐵𝑓_𝐴𝐴()𝐵𝐵=𝐴,𝐶,𝐵−𝐴𝐴(𝐴,𝐶,𝐵)𝐵𝐵. (22) In addition, we have (𝐴,𝐶,𝐵)−𝐴𝐴(𝐴,𝐶,𝐵)=𝐵𝐵_{𝑛−1𝑘=0}𝛼_𝑘^𝑛−1_𝑗=𝑘𝐴𝐴^𝑗−𝑘𝐶𝐵𝐵^𝑗+^𝑛−1_𝑗=𝑘𝐴𝐴^𝑗−𝑘𝐴𝐶𝐵𝐵𝐵^𝑗−^𝑛−1_𝑘=0𝛼_𝑘^𝑛−1_𝑗=𝑘𝐴𝐴^{𝑗−𝑘+1}𝐶𝐵𝐵^𝑗+1+^𝑛−1_𝑗=𝑘𝐴𝐴^{𝑗−𝑘+1}𝐴𝐶𝐵𝐵𝐵^𝑗+1=^𝑛−1_𝑘=0𝛼_𝑘×^𝑛−1_𝑗=𝑘𝐴𝐴^𝑗−𝑘𝐶𝐵𝐵^𝑗−𝐴𝐴^{𝑗−𝑘+1}𝐶𝐵𝐵^𝑗+1+𝐴𝐴^𝑗−𝑘𝐴𝐶𝐵𝐵𝐵^𝑗−𝐴𝐴^{𝑗−𝑘+1}𝐴𝐶𝐵𝐵𝐵^𝑗+1=^𝑛−1_𝑘=0𝛼_𝑘𝐶𝐵𝐵^𝑘−𝐴𝐴^𝑛−𝑘𝐶𝐵𝐵^𝑛+𝐴𝐶𝐵𝐵𝐵^𝑘−𝐴𝐴^𝑛−𝑘𝐴𝐶𝐵𝐵𝐵^𝑛=𝐶𝑓_𝐴𝐴𝐵𝐵−𝑓_𝐴𝐴𝐴𝐴𝐶𝐵𝐵^𝑛+𝐴𝐶𝐵𝑓_𝐴𝐴𝐵𝐵−𝑓_𝐴𝐴𝐴𝐴𝐴𝐶𝐵𝐵𝐵^𝑛=𝐶𝑓_𝐴𝐴𝐵𝐵+𝐴𝐶𝐵𝑓_𝐴𝐴𝐵𝐵. (23)
Combining this with (22) gives 𝑋−𝐴𝐴𝑋𝐵𝐵𝑓_𝐴𝐴𝐵𝐵=𝐶+𝐴𝐶𝐵𝑓_𝐴𝐴𝐵𝐵. (24) Since 𝜂𝛾≠1 for any 𝜂∈𝜆(𝐴𝐴) and 𝛾∈𝜆(𝐵𝐵), the matrix 𝑓_𝐴𝐴(𝐵𝐵) is nonsingular. Hence, it is obtained from (24) that (24) implies (2). With the above two aspects, the conclusion is proved.

The following theorem gives a result on the unique solution of the Kalman-Yakubovich-conjugate matrix equation.

Theorem 3. If 𝜂𝛾≠1 for any 𝜂∈𝜆(𝐴𝐴) and 𝛾∈𝜆(𝐵𝐵), then the unique solution of the matrix equation (2) is 𝑋=(𝐴,𝐶,𝐵)𝑓_𝐴𝐴𝐵𝐵⁻¹, (25) which is a polynomial of matrices 𝐴, 𝐵, and 𝐶.

Proof. Firstly, we assume that the characteristic polynomial of 𝑓_𝐴𝐴(𝐵𝐵) be 𝑓_{𝑓𝐴𝐴(𝐵𝐵)}(𝑠)=∑^𝑝_𝑖=0𝛽_𝑖𝑠^𝑖. Because 𝑓_𝐴𝐴(𝐵𝐵) is nonsingular, it is shown that 𝛽₀≠0. It follows from Cayley-Hamilton’s theorem that 𝑓_{𝑓𝐴𝐴(𝐵𝐵)}𝑓_𝐴𝐴𝐵𝐵=^𝑝_𝑖=1𝛽_𝑖𝑓_𝐴𝐴𝐵𝐵^𝑖+𝛽₀𝐼=0. (26) This relation implies that 𝑓_𝐴𝐴𝐵𝐵^𝑝_𝑖=1𝛽_𝑖𝑓_𝐴𝐴𝐵𝐵^𝑖−1=−𝛽₀𝐼. (27) Therefore, we have 𝑓_𝐴𝐴𝐵𝐵⁻¹1=−𝛽₀^𝑝_𝑖=1𝛽_𝑖𝑓_𝐴𝐴𝐵𝐵^𝑖−1, (28) which is a polynomial of 𝑓_𝐴𝐴(𝐵𝐵). Since 𝑓_𝐴𝐴(𝐵𝐵) is a polynomial of 𝐵𝐵, it is easily known that [𝑓_𝐴𝐴(𝐵𝐵)]⁻¹ is a polynomial of  𝐵𝐵. So we can see that ∏(𝐴,𝐶,𝐵)[𝑓_𝐴𝐴(𝐵𝐵)]⁻¹ is a polynomial of matrices 𝐴, 𝐵, and 𝐶.

Next, we give two equivalent forms of the solution to the matrix equation (2). In order to obtain the unique solution of the matrix equation (2), only the coefficients of characteristic polynomial of 𝐴𝐴 are required. Firstly, the so-called generalized Faddeev-Leverrier algorithm [20] is stated as the following relations: 𝑈_𝑘=𝑈_𝑘−1𝐴𝐴+𝛼_𝑘𝐼,𝑈₀𝛼=𝐼,𝑘=1,2,…,𝑛._𝑘=trace𝑈_𝑘−1𝐴𝐴𝑘,𝛼₀=1,𝑘=1,2,…,𝑛, (29) where 𝛼_𝑖, 𝑖=0,1,2,…,𝑛−1, are the coefficients of the characteristic polynomial of the matrix 𝐴𝐴, and 𝑈_𝑖, 𝑖=0,1,…,𝑛−1, are the coefficient matrices of the adjoint matrix (𝑠𝐼_𝑛−𝐴𝐴)^∗.

So we have the following theorems.

Theorem 4. Given the matrices 𝐴∈𝐶^𝑛×𝑛, 𝐵∈𝐶^𝑝×𝑝, and 𝐶∈𝐶^𝑛×𝑝, let 𝑓_𝐴𝐴(𝑠)=det𝐼−𝑠𝐴𝐴=𝛼_𝑛𝑠^𝑛+⋯+𝛼₁𝑠+𝛼₀,𝛼₀=1,𝐼−𝑠𝐴𝐴^∗=𝑈_𝑛−1𝑠^𝑛−1+⋯+𝑈₁𝑠+𝑈₀. (30)(1)If 𝑋 is a solution to (2), then 𝑋𝑓_𝐴𝐴𝐵𝐵=^𝑛−1_𝑗=0𝑈_𝑗𝐶𝐵𝐵^𝑗+^𝑛−1_𝑗=0𝑈_𝑗𝐴𝐶𝐵𝐵𝐵^𝑗. (31)(2)If the matrix 𝑓_𝐴𝐴(𝐵𝐵) is nonsingularity, then the matrix equation (2) has the unique solution; that is, 𝑋=^𝑛−1_𝑗=0𝑈_𝑗𝐶𝐵𝐵^𝑗+^𝑛−1_𝑗=0𝑈_𝑗𝐴𝐶𝐵𝐵𝐵^𝑗𝑓_𝐴𝐴𝐵𝐵⁻¹. (32)

Proof. By applying (29), we can obtain the following expression: 𝑈₀𝑈=𝐼,₁=𝛼₁𝐼+𝐴𝑈𝐴,₂=𝛼₂𝐼+𝛼₁𝐴𝐴+𝐴𝐴²,⋮𝑈_𝑛−1=𝛼_𝑛−1𝐼+𝛼_𝑛−2𝐴𝐴+⋯+𝐴𝐴^𝑛−1. (33) Meanwhile, the above formula can be stated as 𝑈_𝑗=^𝑗_𝑘=0𝛼_𝑘𝐴𝐴^𝑗−𝑘,𝛼₀=1,𝑗=1,2,…,𝑛−1. (34) Thus, it is easy to know ^𝑛−1_𝑘=0^𝑛−1_𝑗=𝑘𝛼_𝑘𝐴𝐴^𝑗−𝑘𝐶𝐵𝐵^𝑗+^𝑛−1_𝑘=0^𝑛−1_𝑗=𝑘𝛼_𝑘𝐴𝐴^𝑗−𝑘𝐴𝐶𝐵𝐵𝐵^𝑗=^𝑛−1_𝑗=0^𝑗_𝑘=0𝛼_𝑘𝐴𝐴^𝑗−𝑘𝐶𝐵𝐵^𝑗+^𝑛−1_𝑗=0^𝑗_𝑘=0𝛼_𝑘𝐴𝐴^𝑗−𝑘𝐴𝐶𝐵𝐵𝐵^𝑗=^𝑛−1_𝑗=0𝑈_𝑗𝐶𝐵𝐵^𝑗+^𝑛−1_𝑗=0𝑈_𝑗𝐴𝐶𝐵𝐵𝐵^𝑗. (35) Thus, we can easily obtain the conclusions.

Theorem 5. Suppose the matrices 𝐴∈𝐶^𝑛×𝑛, 𝐵∈𝐶^𝑝×𝑝, and 𝐶∈𝐶^𝑛×𝑝.(1)Let 𝑋 be a solution of (2), thus, 𝑋𝑓_𝐴𝐴𝐵𝐵=𝑄_𝑐𝐴𝐴,𝐶,𝑛𝑆_𝑝𝐼,𝐴𝐴𝑄_𝑜𝐵𝐵,𝐼,𝑛+𝑄_𝑐𝐴𝐴,𝐴,𝑛𝑆_𝑝𝐼,𝐴𝐴𝑄_𝑜𝐵𝐵,𝐶𝐵,𝑛. (36)(2)Let 𝜆𝜇≠1 for any 𝜆, 𝜇∈𝜎(𝐴𝐴); thus, the matrix equation (2) has a unique solution 𝑋=𝑄_𝑐𝐴𝐴,𝐶,𝑛𝑆_𝑝𝐼,𝐴𝐴𝑄_𝑜𝐵𝐵,𝐼,𝑛+𝑄_𝑐𝐴𝐴,𝐴,𝑛𝑆_𝑝𝐼,𝐴𝐴𝑄_𝑜𝐵𝐵,×𝑓𝐶𝐵,𝑛_𝐴𝐴𝐵𝐵⁻¹. (37)

Proof. In view of relation (33), it is obvious that 𝑈₀𝐶𝑈₁𝐶⋯𝑈_𝑛−1𝐶=𝑄_𝑐𝐴𝐴,𝐶,𝑛𝑆_𝑝𝐼,𝐴𝐴.𝑈₀𝐴𝑈₁𝐴⋯𝑈_𝑛−1𝐴=𝑄_𝑐𝐴𝐴,𝐴,𝑛𝑆_𝑝𝐼,𝐴𝐴. (38) Then it is easy to obtain that ^𝑛−1_𝑗=0𝑈_𝑗𝐶𝐵𝐵^𝑗=𝑈₀𝐶𝑈₁𝐶⋯𝑈_𝑛−1𝐶𝑄_𝑜𝐵𝐵,𝐼,𝑛=𝑄_𝑐𝐴𝐴,𝐶,𝑛𝑆_𝑝𝐼,𝐴𝐴𝑄_𝑜𝐵𝐵,𝐼,𝑛_{𝑛−1𝑗=0},𝑈_𝑗𝐴𝐶𝐵𝐵𝐵^𝑗=𝑈₀𝐴𝑈₁𝐴⋯𝑈_𝑛−1𝐴𝑄_𝑜𝐵𝐵,𝐶𝐵,𝑛=𝑄_𝑐𝐴𝐴,𝐴,𝑛𝑆_𝑝𝐼,𝐴𝐴𝑄_𝑜𝐵𝐵,𝐶𝐵,𝑛. (39) Combining this with Theorem 4, we complete the proof.

On the basis of the above results, we have the following corollary on the solution of the Stein-conjugate matrix equation 𝑋−𝐴𝑋𝐴^𝑇=𝐶.

Corollary 6. Let the matrices 𝐴∈𝐶^𝑛×𝑛 and 𝐶∈𝐶^𝑛×𝑛 be given. If the matrix 𝐴𝐴 is Schur stable, then the unique solution of the matrix equation 𝑋−𝐴𝑋𝐴^𝑇=𝐶 is expressed as 𝑋=𝐴,𝐶,𝐴^𝑇𝑓_𝐴𝐴𝐴^𝐻𝐴^𝑇−1, (40) which is a polynomial of matrices 𝐴 and 𝐶.

3. Numerical Example

Example 1. Here we give an example for computing the solution to matrix equation 𝑋−𝐴𝑋𝐵=𝐶.
The parametric matrix can be written as follows: ,.𝐴=𝑖0241+𝑖123𝑖0,𝐵=1𝑖−𝑖2𝐶=−1−𝑖−1+4𝑖−8−13𝑖−19+11𝑖−11−𝑖−1+14𝑖 (41) It is easy to check that 𝜂𝛾≠1, for any 𝜂∈𝜆(𝐴𝐴) and 𝛾∈𝜆(𝐵𝐵). So we can see that the above matrix equation has a unique solution.

The following result can be obtained by some simple computation: 𝑓_(𝐴𝐴)(𝑠)=𝑠³−11𝑠²𝑓+43𝑠−401,_(𝐴𝐴)(𝑠)=−401𝑠³+43𝑠²−11𝑠+1. (42) Thus 𝛼₀=1, 𝛼₁=−11, and 𝛼₂=43; in addition, we have 𝑓_(𝐴𝐴)=,𝑓𝐵𝐵11613090𝑖3090𝑖−8109_(𝐴𝐴)𝐵𝐵⁻¹=.−0.0607−0.0231𝑖−0.0231𝑖0.0087 (43) So we have the following matrix expression: ₀(𝐴,𝐶,𝐵)=𝐶+𝐴𝐶𝐵+𝐴𝐴𝐶+𝐴𝐶𝐵+𝐴𝐵𝐵𝐴²𝐶+𝐴𝐶𝐵𝐵𝐵²,₁(𝐴,𝐶,𝐵)=𝐶+𝐴𝐶𝐵𝐵𝐵+𝐴𝐴𝐶+𝐴𝐶𝐵𝐵𝐵²,₂(𝐴,𝐶,𝐵)=𝐶+𝐴𝐶𝐵𝐵𝐵²,𝛼₀₀(𝐴,𝐶,𝐵)+𝛼₁₁(𝐴,𝐶,𝐵)+𝛼₂₂()=.𝐴,𝐶,𝐵1161+6180𝑖−16218+3090𝑖3090+5412𝑖−14289+8109𝑖3090𝑖−8109 (44) Therefore, it follows from Theorem 3 that the unique solution of (2) is 𝑋=𝛼₀₀(𝐴,𝐶,𝐵)+𝛼₁₁(𝐴,𝐶,𝐵)+𝛼₂₂(×𝑓𝐴,𝐶,𝐵)_(𝐴𝐴)𝐵𝐵⁻¹=.1.0000+0.0000𝑖2.0000+0.0000𝑖−0.0000+2.0000𝑖1.0000−1.0000𝑖0.0000𝑖1.0000 (45)

4. Conclusion

The well-known Karm-Yakubovich matrix equation and the generalized discrete Sylvester matrix equation have many important applications in control theory and system theory. As the generalization of the above matrix equations, in this paper we have proposed polynomial solutions to the Karm-Yakubovich-conjugate matrix equation. Different from the other approaches, the approach in the current paper does not require transformation of the coefficient matrices into any canonical form. The solutions are stated as a polynomial of parameters of the matrix equation. All the coefficient matrices are not restricted to be in any canonical form. Meanwhile, an equivalent form of the solutions to the Karm-Yakubovich-conjugate matrix equation has been expressed in terms of controllability matrix associated with 𝐴, 𝐵, and 𝐶 and observability matrix associated with 𝐴 and 𝐵. Such a feature may bring greater convenience and advantages to some problems related to the Karm-Yakubovich-conjugate matrix equation. At the end of the paper, the numerical experiment is done to illustrate the performance of the proposed method. From the discussion in our paper, on one hand, one can observe that the solutions to the Karm-Yakubovich-conjugate matrix equation are crucial as the theoretical basis of the development of many kinds of other matrix equations and deserve further investigation in the future and, on the other hand, as the theoretical generalization of the well-known Karm-Yakubovich matrix equation, it may be helpful for future control applications.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This project is granted financial support from Postdoctoral Science Foundation of China (2013M541900).