A new technique for solving continuous Sylvester-conjugate matrix equations

Abstract

In the present paper, by using of the coefficients of the characteristic polynomial of matrix $B$ and the so-called Leverrier algorithm, the explicit solutions to the Sylvester-conjugate matrix equation $AX - \bar{X} B = C$ (including the Lyapunov-conjugate matrix equation as special case) have been constructed. While one of the explicit solutions is stated as a polynomial of coefficient matrices of the matrix equation, one of the explicit solutions is expressed by the symmetric operator matrix, controllability matrix and observability matrix. Comparing to the existing results, there is no requirement on the coefficient matrices. At the end of this paper, one numerical example is shown to illustrate the effectiveness of the proposed method.

Keywords

Sylvester-conjugate matrix equation Lyapunov-conjugate matrix equation characteristic polynomial explicit solution

Introduction

It is well-known that the Sylvester matrix equation and the Lyapunov matrix equation are very important since they play a fundamental role in some control problems (Jones and Lew, 1982; Xing et al., 2001), and pure mathematics (Huang and Liu, 1997; Roth, 1952; Souza and Bhattacharya, 1981). For example, the continuous-time Lyapunov matrix equation $AX - X A^{T} = C$ and the Sylvester matrix equation $AX - XB = C$ have been used to solve many control problems such as pole assignment, robust pole assignment, eigenstructure assignment and observer design (Souza and Bhattacharya, 1981; Zhou and Duan, 2005). The numerical solutions of the Sylvester and Lyapunov matrix equations have been addressed in a large body of literature. Jameson (1968) proposed an explicit solution in terms of the characteristic polynomial of matrices $A$ and $B$ . Souza and Bhattacharyys (1981) established a closed-form finite series representation of the unique solution. In this solution, some coefficients were closely related to the companion matrices of the characteristic polynomials of matrices $A$ and $B$ . Jones and Lew (1982) established an expression of the solution in terms of the principle idempotents. Deghan and Hajarian (2009) proposed an efficient iterative method for solving the second-order Sylvester matrix equation $EV F^{2} - AVF - CV = BW$ . Gradient-based iterative algorithms (Zhou et al., 2009) were proposed for solving general coupled Sylvester matrix equations arising in linear system theory by using the so-called ‘generalized Sylvester-mapping’ which had some elegant properties. Wu et al. (2012) discussed the closed-form solutions to the generalized Sylvester-conjugate matrix equation, and the proposed solution can provide all the degrees of freedom that are represented by a free parameter matrix. By proposing two operations with respect to the complex matrix, a simple explicit solution to the Sylvester-conjugate matrix equation can be given in finite series form (Wu et al., 2011c, 2011d). Some explicit closed-form solutions of homogeneous and nonhomogeneous Sylvester-conjugate matrix equations have been provided by Wu et al. (2010). By specifying the solutions of the homogeneous Sylvester-conjugate matrix equation, some new expressions of the solutions to the normal Sylvester, normal Sylvester-conjugate and Sylvester equations were given. Wu et al. (2011a) proposed the conjugate product and the Sylvester-conjugate sum for complex polynomial matrices and some important properties for two new operators were proved. Based on these derived results, the authors provided a unified approach to solve the general class of Sylvester-polynomial-conjugate matrix equations, which includes the Yakubovich-conjugate matrix equation as a special case. In addition, Wu et al. (2011b) presented an iterative algorithm to obtain solutions to the coupled Sylvester-conjugate matrix equations. When the considered matrix equation is consistent, a solution can be obtained within finite iteration steps for any initial values in the absence of round-off errors. Another feature of the proposed algorithm is that it can be implemented by using original coefficient matrices, and does not require transformation of the coefficient matrices into any canonical forms. With a real inner product in complex matrix spaces as a tool, an iterative algorithm was presented to solve a class of extended Sylvester-conjugate matrix equations and the proposed algorithm can obtain an exact solution within finite iteration steps for any initial values (Wu et al., 2011e). Wu et al. (2010) proposed a general, complete parametric solution to the nonhomogeneous generalized Sylvester matrix equation $AV + BW = EVF + R$ . One advantage of the proposed solution is that the matrices $F$ and $R$ are in an arbitrary form and can be set undetermined. This may be highly convenient to many problems in descriptor linear systems, such as observer design and model reference control. Li et al. (2009) proposed an iterative algorithm to obtain the linear matrix equation $\sum_{i = 1}^{l} A_{i} X B_{i} = C$ by using the idea of conjugated gradient iteration. The key property of the proposed algorithm is that the iteration converges to the exact solution in finite steps in the absence of run-off errors. Zhou et al. (2010) investigated the problem of parameterizing all solutions to the polynomial Diophantine matrix equation and the generalized Sylverter matrix equation by using the so-called generalized Sylverter mapping, right coprime factorization and Bezout identity associated with certain polynomial matrices. It was shown that the solutions provided can be parameterized as soon as two pairs of polynomial matrices satisfying the right coprime factorization and Bezout identity are obtained. Li et al. (2010) presented two algorithms for solving the minimal norm least-squares solution to the general linear equation $\sum_{i = 1}^{r} A_{i} X B_{i} + \sum_{j = 1}^{s} C_{j} X^{T} D_{j} = E$ based on the gradient-based searching principle and its dual form. Comparing these two methods, the first one minimizes the spectral radius of the iteration matrix and an explicit expression for the optimal step size is obtained. The second method minimizes the square sum of the F-norm of the error matrices produced by the algorithm. The solvability, existence of a unique solution, a closed-form solution and a numerical solution of the matrix equation $X = Af (X) B + C$ with $f (X) = X^{T}, f (X) = \bar{X}$ and $f (X) = X^{H}$ , where $X$ is the unknown matrix, were investigated in Zhou et al. (2011). In addition, numerical solutions are approximated by iterations which are the generalizations of the Smith iteration and accelerated Smith iteration associated with the standard Stein equation. Moreover, Li et al. (2011) considered the iterative solutions of the Lyapunov matrix equation associated with Itô Stochastic systems having Markovian jump parameters. For both discrete-time and continuous-time Lyapunov equations, the convergence rates of the established algorithms were analyzed and compared.

In the present paper, based on the work of Wu et al. (2006), we construct the explicit solutions to the Sylvester-conjugate matrix equation $AX - \bar{X} B = C$ by applying of the coefficients of the characteristic polynomial of matrix $B$ . The proposed approach does not require the coefficient matrices to be in arbitrary canonical form. Similar to Wu et al. (2006), the proposed solution has a neat and elegant form in terms of the symmetric operator matrix, controllability matrix and observability matrix. Based on the proposed solution, an explicit solution to the continuous Lyapunov-conjugate matrix equation is also derived.

As a generalization of the continuous-time matrix equation, the following linear matrix equation

AX - \bar{X} B = C

has received much attention in the literature over the past two decades. Two complex matrices $A$ and $B$ are said to be consimilar if there exists an invertible complex matrix $P$ such that ${\bar{P}}^{- 1} AP = B$ . The authors of Bevis et al. (1987, 1998) and Wu et al. (2008) provided the consistence of the matrix equation $AX - \bar{X} B = C$ by using the consimilarity of the two matrices. Wu et al. (2006) constructed some explicit expressions of the solution to the matrix equation $AX - \bar{X} B = C$ by means of a real representation of a complex matrix. It was shown that there exists a unique solution, if and only if, $A \bar{A}$ and $B \bar{B}$ have no common eigenvalues. It was also shown that the consistence of the matrix equation $AX - \bar{X} B = C$ can be characterized by the consimilarity of two matrices (Horn and Johnson, 1990). The solutions to the matrix equations $X - AXB = C$ and $X - A \bar{X} B = C$ have been expressed in terms of the characteristic polynomial of $A$ (Jiang and Wei, 2003, 2005). Song and Chen (2011) and Song et al. (2012) established the explicit solution of the quaternion matrix equation $XF - AX = C$ and $X - A \tilde{X} F = C$ , where $\tilde{X}$ denotes the $j$ -conjugate of the quaternion matrix. Wang et al. (2008, 2009a, 2009b) investigated the extreme ranks of the solutions to the quaternion matrix equation $AXB = C, AXB + CYD = E$ , respectively, the equivalent canonical form of a matrix triplet over an arbitrary division ring, and a solvable condition for a pair of generalized Sylvester equations and so on. For details, please see the references.

Moreover, other matrix equations such as the coupled Sylvester matrix equations and the transpose matrix equations have also found numerous applications in control theory and have been investigated. For more information see Duan (2004), Song et al. (2011) and Wang et al. (2007) and references therein. Xia et al. (2009, 2008) investigated the stability, state feedback stabilization, state feedback control and static output feedback control for a class of continuous-time singular hybrid systems and a class of discrete-time singular hybrid systems, respectively. Robust Kalman filtering for systems under norm bounded uncertainties in all the system matrices and error covariance constraints is discussed in Xia and Han (2005). Sufficient conditions are given for the existence of such filters in terms of Riccati equations.

In this paper, we investigate the Sylvester-conjugate matrix equation $AX - \bar{X} B = C$ . Our approach differs from that of Wu et al. (2006); by using the coefficient of the characteristic polynomial of matrix $B$ or the so-called ‘Leverrier algorithm’, we construct the explicit solutions to the Sylvester-conjugate matrix equation $AX - \bar{X} B = C$ and the Lyapunov-conjugate matrix equation. While one of the explicit solutions is stated as the polynomial form of coefficient matrices of the matrix equations, one of the explicit solutions is 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.

Throughout this paper, we use $R$ and $C$ to denote the real number field and the complex number field. We use $A^{T}$ , $\bar{A}, A^{H}$ and $adj (A)$ 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 an appropriate dimension identity matrix. Moreover, for $A \in C^{n \times n}$ , $B \in C^{n \times r}$ and $C \in C^{m \times n}$ , we have the following notation

Q_{c} (A, B, n) = [\begin{matrix} B & AB & \dots & A^{n - 1} B \end{matrix}]

Q_{o} (A, C, k) = [\begin{matrix} C 2 \\ CA \\ ⋮ \\ C A^{k - 1} \end{matrix}]

f_{A} (s) = \det (sI - A) = s^{n} + α_{n - 1} s^{n - 1} + \dots + α_{1} s + α_{0}

S_{r} (A) = [\begin{matrix} α_{1} I_{r} & α_{2} I_{r} & α_{3} I_{r} & \dots & α_{n - 1} I_{r} \\ α_{2} I_{r} & α_{3} I_{r} & \dots & I_{r} \\ \dots & \dots \\ α_{n - 1} I_{r} & I_{r} \\ I_{r} \end{matrix}]

In this case, $Q_{c} (A, B, n)$ , $Q_{o} (A, C, k)$ and $S_{r} (A)$ are named the controllability matrix, the observability matrix and the symmetric operator matrix, respectively.

The Sylvester-conjugate matrix equation $AX - \bar{X B} = C$

In this section, the following matrix equation is investigated

AX - \bar{X} B = C

(1)

where $A \in C^{n \times n}$ , $B \in C^{p \times p}$ , $C \in C^{n \times p}$ and $X \in C^{n \times p}$ are unknown.

Lemma 1. Assume that $A \in C^{n \times n}$ , $B \in C^{p \times p}$ and $C \in C^{n \times p}$ , if $X$ is a solution of equation (1). Then for any integer $k \geq 1,$ the following conclusion in equation (2) can be established

\begin{matrix} {(\bar{A} A)}^{k} X - X {(\bar{B} B)}^{k} & = \sum_{i = 0}^{k - 1} {(\bar{A} A)}^{k - 1 - i} (\bar{C} B) {(\bar{B} B)}^{i} \\ + \sum_{i = 0}^{k - 1} {(\bar{A} A)}^{k - 1 - i} (\bar{A} C) {(\bar{B} B)}^{i} \end{matrix}

(2)

Proof. We prove this conclusion by mathematical induction. According to (1), we have

\bar{A} AX - \bar{A} \bar{X} B = \bar{A} C

(3)

and

\bar{A} \bar{X} B - X \bar{B} B = \bar{C} B

(4)

By equations (3) and (4), we have

\bar{A} AX - X \bar{B} B = \bar{A} C + \bar{C} B

(5)

This implies that the relation in equation (2) holds for $k = 1$ . Assume equation (2) holds for $k \leq N$ , i.e.

\begin{matrix} {(\bar{A} A)}^{N} X - X {(\bar{B} B)}^{N} & = \sum_{i = 0}^{N - 1} {(\bar{A} A)}^{N - 1 - i} (\bar{C} B) {(\bar{B} B)}^{i} \\ + \sum_{i = 0}^{N - 1} {(\bar{A} A)}^{N - 1 - i} (\bar{A} C) {(\bar{B} B)}^{i} \end{matrix}

(6)

Pre-multiplying by $\bar{A} A$ and post-multiplying by $\bar{B} B$ in the both sides of equation (6), we have

\begin{matrix} {(\bar{A} A)}^{N + 1} X - \bar{A} AX {(\bar{B} B)}^{N} = & (\bar{A} A) \sum_{i = 0}^{N - 1} {(\bar{A} A)}^{N - 1 - i} (\bar{C} B) {(\bar{B} B)}^{i} \\ + (\bar{A} A) \sum_{i = 0}^{N - 1} {(\bar{A} A)}^{N - 1 - i} (\bar{A} C) {(\bar{B} B)}^{i} \end{matrix}

(7)

and

\begin{matrix} {(\bar{A} A)}^{N} X \bar{B} B - X {(\bar{B} B)}^{N + 1} = & (\bar{A} A) \sum_{i = 0}^{N - 1} {(\bar{A} A)}^{N - 1 - i} (\bar{C} B) {(\bar{B} B)}^{i + 1} \\ + (\bar{A} A) \sum_{i = 0}^{N - 1} {(\bar{A} A)}^{N - 1 - i} (\bar{A} C) {(\bar{B} B)}^{i + 1} \end{matrix}

(8)

Combining equation (7) with equation (8), we can obtain

\begin{matrix} {(\bar{A} A)}^{N + 1} X - X {(\bar{B} B)}^{N + 1} + {(\bar{A} A)}^{N} X \bar{B} B - \bar{A} AX {(\bar{B} B)}^{N} \\ = \sum_{i = 0}^{N - 1} {(\bar{A} A)}^{N - i} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{i} \\ + (\bar{A} A) \sum_{i = 0}^{N - 1} {(\bar{A} A)}^{N - 2 - i} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{i} (\bar{B} B) \end{matrix}

(9)

Therefore, we have

\begin{matrix} {(\bar{A} A)}^{N + 1} X - X {(\bar{B} B)}^{N + 1} \\ = \sum_{i = 0}^{N - 1} {(\bar{A} A)}^{N - i} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{i} \\ + \sum_{i = 0}^{N - 1} {(\bar{A} A)}^{N - 1 - i} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{i + 1} + \bar{A} AX {(\bar{B} B)}^{N} - {(\bar{A} A)}^{N} X (\bar{B} B) \\ = \sum_{i = 0}^{N - 1} {(\bar{A} A)}^{N - i} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{i} + \sum_{i = 0}^{N - 1} {(\bar{A} A)}^{N - 1 - i} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{i + 1} \\ - \bar{A} A [{(\bar{A} A)}^{N - 1} X - X {(\bar{B} B)}^{N - 1}] (\bar{B} B) \\ = \sum_{i = 0}^{N - 1} {(\bar{A} A)}^{N - i} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{i} + \sum_{i = 0}^{N - 1} {(\bar{A} A)}^{N - 1 - i} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{i + 1} \\ - \bar{A} A \sum_{i = 0}^{N - 2} {(\bar{A} A)}^{N - 2 - i} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{i} (\bar{B} B) \\ = \sum_{i = 0}^{N - 1} {(\bar{A} A)}^{N - i} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{i} + \sum_{i = 0}^{N - 1} {(\bar{A} A)}^{N - 1 - i} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{i + 1} \\ - \sum_{i = 0}^{N - 2} {(\bar{A} A)}^{N - 1 - i} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{i + 1} \\ = \sum_{i = 0}^{N - 1} {(\bar{A} A)}^{N - i} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{i} + (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{N} \\ = \sum_{i = 0}^{N} {(\bar{A} A)}^{N - i} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{i} \end{matrix}

(10)

This implies that the relation in equation (2) holds for $k = N$ . So the conclusion is true.

Define $\nabla_{k} (A, C, B) = \sum_{i = 0}^{k} {(\bar{A} A)}^{k - i} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{i},$ then the equality in equation (2) can be compactly written as

{(\bar{A} A)}^{k} X - X {(\bar{B} B)}^{k} = \nabla_{k - 1} (A, C, B)

(11)

Let

\begin{matrix} p (s) & = \det (sI - \bar{A} A) = \sum_{i = 0}^{n} α_{i} s^{i} \\ = s^{n} + α_{n - 1} s^{n - 1} + \dots + α_{1} s + α_{0} \end{matrix}

(12)

and

\begin{matrix} q (s) = & \det (sI - \bar{B} B) = \sum_{i = 0}^{n} β_{i} s^{i} = s^{n} \\ + β_{n - 1} s^{n - 1} + \dots + β_{1} s + β_{0} \end{matrix}

(13)

From equations (11), (12) and (13), it follows that

\begin{matrix} \sum_{k = 1}^{n} β_{k} [{(A \bar{A})}^{k} X - X {(\bar{B} B)}^{k}] + β_{0} (X - X) \\ = [\sum_{k = 1}^{n} β_{k} {(A \bar{A})}^{k} X + β_{0} X] - [\sum_{k = 1}^{n} β_{k} X {(\bar{B} B)}^{k} + β_{0} X] \\ = [\sum_{k = 1}^{n} β_{k} {(A \bar{A})}^{k} + β_{0} I] X - X [\sum_{k = 1}^{n} β_{k} {(\bar{B} B)}^{k} + β_{0} I] \\ = q (\bar{A} A) X - Xq (\bar{B} B) \\ = q (\bar{A} A) X \end{matrix}

On the other hand

\sum_{k = 1}^{n} β_{k} [{(\bar{A} A)}^{k} X - X {(\bar{B} B)}^{k}] = \sum_{k = 1}^{n} β_{k} \nabla_{k - 1} (A, C, B)

Now, we define

\nabla (A, C, B) = \sum_{k = 1}^{n} β_{k} \nabla_{k - 1} (A, C, B)

Obviously, $\nabla (A, C, B)$ is a polynomial of the matrices $A$ , $B$ and $C$ . This polynomial is defined by the coefficient matrices and the characteristic polynomial of $\bar{A} A$ . This implies that for each Sylvester-conjugate equation of the form of equation (1), there is a uniquely determined polynomial $\nabla (A, C, B)$ of its coefficients matrices. Thus, we obtain the following equation

q (\bar{A} A) X = \nabla (A, C, B)

(14)

Theorem 1. If $λ (\bar{A} A) ⋂_{λ} (\bar{B} B) = ϕ$ , then equation (14) is equivalent to equation (1).

Proof. Based on the argument above, it is derived that equation (1) implies equation (14). Now we prove that equation (14) implies equation (1) when $λ (\bar{A} A) ⋂_{λ} (\bar{B} B) = ϕ$ . Suppose that $X$ is a solution of equation (14), the following relation can be obtained

\begin{matrix} (\bar{A} A) [q (\bar{A} A) X] - [q (\bar{A} A) X] (\bar{B} B) \\ = q (\bar{A} A) (\bar{A} AX - X \bar{B} B) \\ = (\bar{A} A) \nabla (A, C, B) - \nabla (A, C, B) (\bar{B} B) \end{matrix}

(15)

Moreover, we have

\begin{array}{l} (\bar{A} A) \nabla (A, C, B) - \nabla (A, C, B) (\bar{B} B) \\ \begin{array}{l} = (\bar{A} A) \sum_{i = 0}^{n} β_{i} \sum_{j = 0}^{i - 1} {(\bar{A} A)}^{i - 1 - j} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{j} \\ - \sum_{i = 0}^{n} β_{i} \sum_{j = 0}^{i - 1} {(\bar{A} A)}^{i - 1 - j} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{j} (\bar{B} B) \end{array} \\ \begin{array}{l} = \sum_{i = 0}^{n} β_{i} \sum_{j = 0}^{i - 1} {(\bar{A} A)}^{i - j} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{j} \\ - \sum_{i = 0}^{n} β_{i} \sum_{j = 0}^{i - 1} {(\bar{A} A)}^{i - 1 - j} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{j + 1} \end{array} \\ = \sum_{i = 0}^{n} β_{i} [\sum_{j = 0}^{i - 1} {(\bar{A} A)}^{i - j} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{j} - \sum_{j = 0}^{i - 1} {(\bar{A} A)}^{i - 1 - j} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{j + 1}] \\ = \sum_{i = 0}^{n} β_{i} [{(\bar{A} A)}^{i} (\bar{C} B + \bar{A} C) - (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{i}] \\ = \sum_{i = 0}^{n} β_{i} {(\bar{A} A)}^{i} (\bar{C} B + \bar{A} C) - \sum_{i = 0}^{n} β_{i} (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{i} \\ = q (\bar{A} A) (\bar{C} B + \bar{A} C) - (\bar{C} B + \bar{A} C) q (\bar{B} B) \\ = q (\bar{A} A) (\bar{C} B + \bar{A} C) \end{array}

That is

q (\bar{A} A) (\bar{A} AX - X \bar{B} B) = q (\bar{A} A) (\bar{C} B + \bar{A} C)

Since $q (s)$ is the characteristic polynomial of $\bar{B} B$ and $λ (\bar{A} A) ⋂ λ (\bar{B} B) = ϕ$ , $q (\bar{A} A)$ is nonsingular. With the above two aspects, the conclusion is proved.▪

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

Theorem 2. If $λ (\bar{A} A) ⋂ λ (\bar{B} B) = ϕ$ , then the unique solution of the matrix equation (1) is

X = {[q (\bar{A} A)]}^{- 1} \nabla (A, C, B)

which is a polynomial of matrices $A$ , $B$ and $C$ .

Proof. We only need to show that ${[q (\bar{A} A)]}^{- 1} \nabla (A, C, B)$ is a polynomial of matrices $A$ , $B$ and $C$ . In other words, we need to show ${[q (\bar{A} A)]}^{- 1}$ is a polynomial of $\bar{A} A$ . Let the characteristic polynomial of $q (\bar{A} A)$ be $f (s) = \sum_{i = 0}^{n} γ_{i} s^{i}$ where $γ_{n} = 1$ . Since $q (\bar{A} A)$ is nonsingular, it is shown that $γ_{0} \neq 0 .$ According to the Cayley–Hamilton theorem, we have

f (q (\bar{A} A)) = \sum_{i = 1}^{n} γ_{i} {[q (\bar{A} A)]}^{i} + γ_{0} I = 0

This relation implies that

q (\bar{A} A) \sum_{i = 1}^{n} γ_{i} {[q (\bar{A} A)]}^{i - 1} = - γ_{0} I

Therefore, we have

{[q (\bar{A} A)]}^{- 1} = - \frac{1}{γ_{0}} \sum_{i = 1}^{n} γ_{i} {[q (\bar{A} A)]}^{i - 1}

which is a polynomial of $q (\bar{A} A)$ . Since $q (\bar{A} A)$ is a polynomial of $\bar{A} A$ , it is easily known that ${[q (\bar{A} A)]}^{- 1}$ is a polynomial of $\bar{A} A$ . So we can see that ${[q (\bar{A} A)]}^{- 1} \nabla (A, C, B)$ is a polynomial of matrices $A$ , $B$ and $C$ .

In the following section, we propose two equivalent forms of the solution to the matrix equation (1). In order to obtain the unique solution of the matrix equation (1), only the coefficients of the characteristic polynomial of $\bar{B} B$ are required. First of all, the so-called ‘generalized Faddeev–Leverrier algorithm’ (Hanzon, 1996) is stated as the following relations

{\begin{matrix} T_{n - k} = T_{n - k + 1} (\bar{B} B) + β_{n - k + 1} I_{n}, & T_{n} = 0, k = 1, 2, \dots, n \\ β_{n - k} = \frac{trace (T_{n - k} \bar{B} B)}{k}, & β_{n} = 1, k = 1, 2, \dots, n \end{matrix}

(16)

where $β_{i}, i = 0, 1, 2, \dots, n - 1$ , are the coefficients of the characteristic polynomial of the matrix $\bar{B} B$ , and $T_{i}, i = 0, 1, \dots, n - 1$ , are the coefficient matrices of the adjoint matrix $adj (sI - \bar{B} B)$ .

Thus we have the following theorems.

Theorem 3. Given the matrices $A \in C^{n \times n}$ , $B \in C^{p \times p}$ , $C \in C^{n \times p}$ , let

q (s) = \det (sI - \bar{B} B) = β_{p} s^{p} + \dots + β_{1} s + β_{0}, β_{p} = 1

and

adj (sI - \bar{B} B) = T_{p - 1} s^{p - 1} + \dots + T_{1} s + T_{0}

If $X$ be a solution to the matrix equation (1), then

q (\bar{A} A) X = \sum_{i = 0}^{n - 1} {(\bar{A} A)}^{k - i - 1} \bar{C} B T_{i} + \sum_{i = 0}^{n - 1} {(\bar{A} A)}^{k - i - 1} \bar{A} C T_{i}

(17)

If $X$ be the unique solution of matrix equation (1), then

X = {[q (\bar{A} A)]}^{- 1} [\sum_{i = 0}^{n - 1} {(\bar{A} A)}^{k - i - 1} \bar{C} B T_{i} + \sum_{i = 0}^{n - 1} {(\bar{A} A)}^{k - i - 1} \bar{A} C T_{i}]

Proof. By applying equation (16), we can obtain the following expression

{\begin{matrix} T_{0} = β_{1} I + β_{2} \bar{B} B + \dots + β_{p - 1} {(\bar{B} B)}^{p - 2} + {(\bar{B} B)}^{p - 1} \\ ⋮ \\ T_{p - 2} = β_{p - 1} I + \bar{B} B \\ T_{p - 1} = I \end{matrix}

(18)

Meanwhile, the above formula can be stated as

T_{i} = \sum_{k = i + 1}^{p} β_{k} {(\bar{B} B)}^{k - i - 1}, β_{p} = 1, i = 1, 2, \dots, p - 1

Thus, it is easy to know

\begin{matrix} \sum_{k = 1}^{n} \sum_{i = 0}^{k - 1} β_{k} {(\bar{A} A)}^{k - i - 1} \bar{C} B {(\bar{B} B)}^{i} + \sum_{k = 1}^{n} \sum_{i = 0}^{k - 1} β_{k} {(\bar{A} A)}^{k - i - 1} \bar{A} C {(\bar{B} B)}^{i} \\ = \sum_{k = 1}^{n} \sum_{i = 0}^{k - 1} β_{k} {(\bar{A} A)}^{i} \bar{C} B {(\bar{B} B)}^{k - i - 1} + \sum_{k = 1}^{n} \sum_{i = 0}^{k - 1} β_{k} {(\bar{A} A)}^{i} \bar{A} C {(\bar{B} B)}^{k - i - 1} \\ = \sum_{i = 0}^{n - 1} \sum_{k = i + 1}^{n} β_{k} {(\bar{A} A)}^{i} \bar{C} B {(\bar{B} B)}^{k - i - 1} + \sum_{i = 0}^{n - 1} \sum_{k = i + 1}^{n} β_{k} {(\bar{A} A)}^{i} \bar{A} C {(\bar{B} B)}^{k - i - 1} \\ = \sum_{i = 0}^{n - 1} {(\bar{A} A)}^{i} \bar{C} B (\sum_{k = i + 1}^{n} β_{k} {(\bar{B} B)}^{k - i - 1}) \\ + \sum_{i = 0}^{n - 1} {(\bar{A} A)}^{i} \bar{A} C (\sum_{k = i + 1}^{n} β_{k} {(\bar{B} B)}^{k - i - 1}) \\ = \sum_{i = 0}^{n - 1} {(\bar{A} A)}^{i} \bar{C} B T_{i} + \sum_{i = 0}^{n - 1} {(\bar{A} A)}^{i} \bar{A} C T_{i} \end{matrix}

Thus, we can easily obtain the conclusions.

Theorem 4. Suppose the matrices $A \in C^{n \times n}$ , $B \in C^{p \times p}$ , $C \in C^{n \times p}$ .

If $X$ be a solution of the matrix equation (1), then

\begin{matrix} q (\bar{A} A) X = & Q_{c} (\bar{A} A, \bar{C}, n) S_{n} (\bar{B} B) Q_{o} (\bar{B} B, B, n) \\ + Q_{c} (\bar{A} A, \bar{A}, n) S_{n} (\bar{B} B) Q_{o} (\bar{B} B, C, p) \end{matrix}

If $λ (\bar{A} A) ⋂ λ (\bar{B} B) = ϕ$ , thus, the matrix equation (1) has a unique solution

\begin{matrix} X = {[q (\bar{A} A)]}^{- 1} [Q_{c} (\bar{A} A, \bar{C}, n) S_{n} (\bar{B} B) Q_{o} (\bar{B} B, B, n) + Q_{c} (\bar{A} A, \bar{A}, n) S_{n} (\bar{B} B) Q_{o} (\bar{B} B, C, p) \end{matrix}]

Proof. In view of the relation in equation (18), it is obvious that

[\begin{matrix} B T_{0} C & B T_{1} C & \dots & B T_{p - 1} C \end{matrix}] = S_{n} (\bar{B} B) Q_{o} (\bar{B} B, B, n)

and

[\begin{matrix} C T_{0} & C T_{1} & \dots & C T_{p - 1} \end{matrix}] = S_{n} (\bar{B} B) Q_{o} (\bar{B} B, C, p)

Then it is easy to obtain that

\begin{matrix} \sum_{i = 0}^{n - 1} {(\bar{A} A)}^{k - i - 1} \bar{C} B T_{i} & = Q_{c} (\bar{A} A, \bar{C}, n) [\begin{matrix} B T_{0} & B T_{1} & \dots & B T_{n - 1} \end{matrix}] \\ = Q_{c} (\bar{A} A, \bar{C}, n) S_{n} (\bar{B} B) Q_{o} (\bar{B} B, B, n) \end{matrix}

and

\begin{matrix} \sum_{i = 0}^{n - 1} {(\bar{A} A)}^{k - i - 1} \bar{A} C T_{i} & = Q_{c} (\bar{A} A, \bar{A}, n) [\begin{matrix} C T_{0} & C T_{1} & \dots & C T_{n - 1} \end{matrix}] \\ = Q_{c} (\bar{A} A, \bar{A}, n) S_{n} (\bar{B} B) Q_{o} (\bar{B} B, C, p) \end{matrix}

Combining this with Theorem 3, we complete the proof.

On the basis of the above results, we have the following corollary on the solution of the Lyapunov-conjugate matrix equation $AX - \bar{X} A^{T} = C$ .

Corollary 1. Given matrices $A \in C^{n \times n}$ and $C \in C^{n \times n}$ . Let

\begin{matrix} f (s) = & \det (sI - A^{H} A^{T}) = δ_{n} s^{n} \\ + δ_{n - 1} s^{n - 1} + \dots + δ_{1} s + δ_{0}, δ_{n} = 1 \end{matrix}

and

adj (sI - A^{H} A^{T}) = M_{n - 1} s^{n - 1} + \dots + M_{1} s + M_{0}

If the matrix $A \bar{A}$ is Schur stable, then the unique solution of matrix equation $AX - \bar{X} A^{T} = C$ is expressed as

X = {[f (\bar{A} A)]}^{- 1} \nabla (A, C, A^{T})

which is a polynomial of matrices $A$ and $C$ .

Corollary 2. Given matrices $A \in C^{n \times n}$ and $C \in C^{n \times n}$ . Let

\begin{matrix} f (s) = & \det (sI - A^{H} A^{T}) = δ_{n} s^{n} \\ + δ_{n - 1} s^{n - 1} + \dots + δ_{1} s + δ_{0}, δ_{n} = 1 \end{matrix}

and

adj (sI - A^{H} A^{T}) = M_{n - 1} s^{n - 1} + \dots + M_{1} s + M_{0}

If $X$ be a solution to the Lyapunov-conjugate matrix equation $AX - \bar{X} A^{T} = C$ , then

\begin{matrix} f (\bar{A} A) X = & Q_{c} (A \bar{A}, \bar{C}, n) S_{n} (A^{H} A^{T}) Q_{n} (A^{H} A^{T}, A^{T}, n) \\ + Q_{c} (A \bar{A}, \bar{A}, n) S_{n} (A^{H} A^{T}) Q_{n} (A^{H} A^{T}, C, n) \end{matrix}

If the matrix $A^{2}$ is Schur stable, then the unique solution to Lyapunov-conjugate matrix equation $AX - \bar{X} A^{T} = C$ can be characterized by

\begin{matrix} X = {[f (\bar{A} A)]}^{- 1} [Q_{c} (A \bar{A}, \bar{C}, n) S_{n} (A^{H} A^{T}) Q_{o} (A^{H} A^{T}, A^{T}, n) + Q_{c} (A \bar{A}, \bar{A}, n) S_{n} (A^{H} A^{T}) Q_{o} (A^{H} A^{T}, C, n) \end{matrix}]

Corollary 3. Given matrices $A \in C^{n \times n}$ and $C \in C^{n \times n}$ . Let

\begin{matrix} f (s) = & \det (sI - A^{H} A^{T}) = δ_{n} s^{n} \\ + δ_{n - 1} s^{n - 1} + \dots + δ_{1} s + δ_{0}, δ_{n} = 1 \end{matrix}

and

adj (sI - A^{H} A^{T}) = M_{n - 1} s^{n - 1} + \dots + M_{1} s + M_{0}

If $X$ be a solution to Lyapunov-conjugate matrix equation $AX - \bar{X} A^{T} = C$ , then

f (\bar{A} A) X = \sum_{i = 0}^{n - 1} {(\bar{A} A)}^{k - i - 1} \bar{C} A^{T} M_{i} + \sum_{i = 0}^{n - 1} {(\bar{A} A)}^{k - i - 1} \bar{A} C M_{i}

If $X$ is the unique solution to Lyapunov-conjugate matrix equation $AX - \bar{X} A^{T} = C$ , it can be characterized by

X = {[f (\bar{A} A)]}^{- 1} [\sum_{i = 0}^{n - 1} {(\bar{A} A)}^{k - i - 1} \bar{C} A^{T} M_{i} + \sum_{i = 0}^{n - 1} {(\bar{A} A)}^{k - i - 1} \bar{A} C M_{i}]

Numerical example

Example 1. Here we give an example for computing the solution to the matrix equation $AX - \bar{X} B = C$

with the following parametric matrices

\begin{matrix} A = [\begin{matrix} 3 & i & - i \\ 3 & 1 - i & 0 \\ 0 & i & 4 + 2 i \end{matrix}], B = [\begin{matrix} 1 - i & 1 + i & 2 \\ 3 & 1 & 0 \\ - 2 i & 2 - i & 0 \end{matrix}], \\ C = [\begin{matrix} 8 + 3 i & - 2 + i & - 1 + 2 i \\ - 2 - 2 i & - 1 - 4 i & - 4 + i \\ - 3 + 13 i & 3 - i & 2 + 12 i \end{matrix}] \end{matrix}

It is easy to check $λ (\bar{A} A) ⋂ λ (\bar{B} B) = ϕ$ . So we can see the above matrix equation has a unique solution.

The following result can be obtained by some simple computation

q (s) = \det (sI - \bar{B} B) = s^{3} - 9 s^{2} + 24 s - 148

Thus $β_{0} = - 148, β_{1} = 24, β_{2} = - 9, β_{3} = 1$ , in addition, we have

q (A \bar{A}) = [\begin{matrix} - 274 + 63 i & - 112 - 108 i & - 596 + 661 i \\ - 186 + 3 i & - 149 - 129 i & - 396 - 597 i \\ - 132 - 333 i & 177 + 927 i & 5149 + 66 i \end{matrix}]

\begin{matrix} {[q (\bar{A} A)]}^{- 1} = \\ [\begin{matrix} - 0.0028 - 0.0004 i & - 0.0003 + 0.0003 i & - 0.0004 + 0.0003 i \\ 0.0018 + 0.0005 i & - 0.0038 + 0.0004 i & - 0.0001 - 0.0006 i \\ - 0.0005 i & 0.0002 + 0.0007 i & 0.0001 \end{matrix}] \end{matrix}

According to the following matrix equation expression

\begin{matrix} \nabla_{0} (A, C, B) & = \bar{C} B + \bar{A} C \\ = [\begin{matrix} 4 - 4 i & - 4 + 8 i & 2 + 6 i \\ 19 + 29 i & - 17 + 4 i & - 12 + 7 i \\ - 19 + 49 i & 11 - 50 i & 27 + 22 i \end{matrix}] \end{matrix}

\begin{matrix} \nabla_{1} (A, C, B) & = \bar{A} A (\bar{C} B + \bar{A} C) + (\bar{C} B + \bar{A} C) \bar{B} B \\ = [\begin{matrix} - 105 - 115 i & 155 i & - 62 + 49 i \\ 235 + 219 i & - 245 + 63 i & - 123 + 47 i \\ - 190 + 1410 i & 297 - 857 i & 367 + 273 i \end{matrix}] \end{matrix}

\begin{matrix} \nabla_{2} (A, C, B) & = {(\bar{A} A)}^{2} (\bar{C} B + \bar{A} C) + \bar{A} A (\bar{C} B + \bar{A} C) \bar{B} B + (\bar{C} B + \bar{A} C) {(\bar{B} B)}^{2} \\ = [\begin{matrix} - 3630 - 2768 i & 710 + 2948 i & - 1926 + 88 i \\ 2837 - 19 i & - 3536 + 537 i & - 621 - 924 i \\ - 303 + 28305 i & 7867 - 15818 i & 8071 + 7012 i \end{matrix}] \end{matrix}

\begin{matrix} β_{1} \nabla_{0} (A, C, B) + β_{2} \nabla_{1} (A, C, B) + β_{3} \nabla_{2} (A, C, B) \\ = [\begin{matrix} - 2589 - 1829 i & 614 + 1745 i & - 1320 - 209 i \\ 1178 - 1294 i & - 1739 - 66 i & 198 - 1179 i \\ 951 + 16791 i & 5458 - 9305 i & 5416 + 5083 i \end{matrix}] \end{matrix}

Therefore, it follows from Theorem 2 that the unique solution of equation (1) is

\begin{matrix} X & = {[q (A \bar{A})]}^{- 1} [β_{1} \nabla_{0} (A, C, B) + β_{2} \nabla_{1} (A, C, B) + β_{3} \nabla_{2} (A, C, B)] \\ = [\begin{matrix} 1.0000 & 0.0000 & 1.0000 i \\ 2.0000 - 1.0000 i & 1.0000 & - 0.0000 \\ 3.0000 i & 1.0000 - 2.0000 i & 1.0000 + 1.0000 i \end{matrix}] \end{matrix}

Conclusions

In this paper we have proposed a new method for obtaining the expression of the solutions to the Sylvester-conjugate matrix equation $AX - \bar{X} B = C$ . Different from the approach of Bevis et al. (1998), the approaches in the current paper do not require transformation of the coefficient matrices into any canonical forms. The solutions are constructed for the Sylvester-conjugate matrix equation by using the coefficients of the characteristic polynomial of matrix $B$ and the so-called Leverrier algorithm, and can be obtained by using the original coefficient matrices. One of the explicit solutions is stated as a polynomial of coefficient matrices of the matrix equation. An equivalent form of the solutions to the Sylvester-conjugate matrix equations has been expressed in terms of the controllability matrix associated with $A$ and $C$ , and observability matrices associated with $B$ and $C$ . At the end of the paper, a numerical experiment was carried out to illustrate the performance of the proposed method. From the discussion in the paper, one can observe that the solutions to the Sylvester-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.

Footnotes

Funding

This project was supported the NNSF of China (grant numbers 61174141, 11171226 and 11301247), the Postdoctoral Science Foundation of China (grant number 2013M541900), the Research Awards for Young and Middle-Aged Scientists of Shandong Province (grant numbers BS2011SF009 and BS2011DX019), and the Excellent Youth Foundation of Shandong’s Natural Scientific Committee (grant number JQ201219).

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A new technique for solving continuous Sylvester-conjugate matrix equations

Abstract

Keywords

Introduction

The Sylvester-conjugate matrix equation AX − X B ¯ = C

Numerical example

Conclusions

Footnotes

Funding

References

The Sylvester-conjugate matrix equation $AX - \bar{X B} = C$