The least-squares solutions of the matrix equation $A^{\ast}XB+B^{\ast}X^{\ast}A = D$ and its optimal approximation

1.   Introduction

Throughout this paper, the complex $m\times n$ matrix space is denoted by $\mathbb{C}^{m\times n}$ and the set of all $n\times n$ unitary matrices is denoted by $\mathbb{UC}^{n \times n}$. The conjugate transpose and the Frobenius norm of a complex matrix $A$ are denoted by $A^{\ast}\triangleq \bar{A}^{\top}$ and $\| A \|,$ respectively. The Hermitian (skew-Hermitian) matrix $A$ is denoted by $A = A^{\ast} (A = -A^{\ast})$. The identity matrix of size $n$ is represented by $I_n$. For matrices $A = (\alpha_{ij})\in\mathbb{C}^{m \times n}, B = (\beta_{ij})\in\mathbb{C}^{m \times n},$ $A\ast B$ is used to define the Hadamard product of $A$ and $B$, that is, $A\ast B = (\alpha_{ij} \beta_{ij}) \in\mathbb{C}^{m \times n}$.

It is known that the matrix equation

plays an important role in automatic control. In 1991, Yasuda and Skelton ^[1] studied the assigning controllability and observability Gramians in feedback control by (1.1). In 1994, Fujioka and Hara ^[2] considered (1.1) in the context of studying state covariance assignment problem with measurement noise. Owing to its important applications, there has been an increased interest in solving (1.1). In 1980, Baksalary and Kala ^[3] established the solvability conditions and the representation of the general solution to the matrix equation $AXB+CYD = E$. In 1987, Chu ^[4] considered the compatibility of $AXB+CYD = E$ by using the generalized singular value decomposition (GSVD), and provided the least norm solution when the solution exists. After that, Chu ^[5] provided the solvability conditions of (1.1) by using {the} GSVD. In addition, some iterative methods ^[6,7,8,9] are also used to solve such matrix equations. There is no doubt that the researching of the least-squares solutions to this kind of matrix equation should also be significant and interesting. In 1998, Xu et al. ^[10] provided the least-squares Hermitian (skew-Hermitian) solutions of $AXA^{\mathrm{H}}+CYC^{\mathrm{H}} = F$ by using the canonical correlation decomposition (CCD). In 2006, Liao et al. ^[11] considered the least-squares solution with the minimum norm of $AXB^{\mathrm{H}}+CYD^{\mathrm{H}} = E$ by {the} CCD and GSVD. Yuan et al. ^[12,13] considered the least-squares solutions with some constraints of the matrix equation $AXB+CXD = E$. Besides, other scholars ^[15,16] also have studied the least-squares problems of $AXB+CXD = E$. However, solving the least-squares solutions of $(1.1)$ seems to be rarely considered in the literatures. Recently, Yuan ^[17] proposed the least-squares solutions to the matrix equation $A^{\top}XB-B^{\top}X^{\top}A = D$ by applying the canonical correlation decomposition of $[A^{\top}, B^{\top}]$. Subsequently, Yuan ^[18,19] proposed the minimum norm solution of (1.1) by taking advantage of the generalized singular value decomposition of the matrix pair $[A^{\ast}, B^{\ast}]$, and provided the least-squares solution with the minimum norm of (1.1) by using the normal equation and singular value decompositions. Motivated by the work above, it occurred to us that can the least-squares solutions to (1.1) be derived in a similar way? The answer is affirmative. In this paper, we will consider the least-squares solutions of (1.1) and the associated weighted optimal approximation problem by utilizing the canonical correlation decomposition of a pair of matrices, which can be mathematically formulated as follows.

Problem I. Given $A \in\mathbb{C}^{n\times m}, B \in\mathbb{C}^{p\times m}, D \in\mathbb{C}^{m\times m}$ with $D = D^{\ast}$. Find $X \in\mathbb{C}^{n\times p}$ such that

Problem II. Given $F \in {\mathbb{C}}^{n\times p}$, find $\hat{X}\in \mathcal{S}_{E}$ such that

where $\| \cdot \|_{W}$ is the weighted norm will be defined below, $\mathcal{S}_{E}$ is the solution set of Problem I.

By using the canonical correlation decomposition, the explicit expression of the least-squares solutions to Problem I is derived. Also, the expression of the corresponding optimal approximation solution under a weighted Frobenius norm sense to Problem II is deduced. Further, numerical examples are provided to verify the correctness of our results.

2.   Solution to Problem I

In order to solve Problem I, the following lemmas are needed.

Lemma 2.1. ^[10] Let $J_{1}\in \mathbb{C}^{m\times n}, J_{2} \in \mathbb{C}^{n\times m}, \; C_{A} = \mathit{\mbox{diag}}(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{m}), S_{A} = \mathit{\mbox{diag}}(\beta_{1}, \beta_{2}, \cdots, \beta_{m})$ with $\alpha_{i} > 0, \beta_{i} > 0,$ and $\alpha_{i}^{2}+\beta_{i}^{2} = 1, (i = 1, 2, \cdots, m).$ Then the following minimization problem with respect to $X\in \mathbb{C}^{m\times n}:$

holds if and only if $X$ can be expressed as $X = C_{A}J_{1}+ S_{A}J_{2}^{\ast}.$

Lemma 2.2. Let $J_{1}, J_{2}\in \mathbb{C}^{m\times m}$ with $J_{1} = J_{1}^{\ast}$, $C_{A} = \mathit{\mbox{diag}}(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{m}), S_{A} = \mathit{\mbox{diag}}(\beta_{1}, \beta_{2}, \cdots, \beta_{m})$ with $\alpha_{i} > 0, \beta_{i} > 0,$ and $\alpha_{i}^{2}+\beta_{i}^{2} = 1, (i = 1, 2, \cdots, m).$ Then the following minimization problem with respect to $X\in \mathbb{C}^{m\times m}:$

holds if and only if $X$ can be expressed as

Proof. For $X = (x_{ij})\in \mathbb{C}^{m\times m}, \ J_{l} = (J_{ij}^{(l)})\in \mathbb{C}^{m\times m}, \ (l = 1, 2)$, we have

Clearly, $\phi_{2}$ is a continuously differentiable function of $2m^2$ variables of $\mbox{Re}(x_{ij})$, $\mbox{Im}(x_{ij}), \; (i, j = 1, 2, \; \cdots, m).$ The function of $x_{ij}$ is

According to the necessary condition of function which is minimizing at a point and $J_{ij}^{(1)} = J_{ji}^{(1)}$, we obtain the following expression:

Then the equation of $(2.1)$ follows from $(2.2)$.

Lemma 2.3. Let $J \in \mathbb{C}^{m\times m}, \; C_{A} = \mathit{\mbox{diag}}(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{m})$ with $0 < \alpha_{i} < 1, \ (i = 1, 2, \cdots, m).$ Then the following equation with respect to $X\in \mathbb{C}^{m\times m}:$

holds if and only if $X$ can be expressed as

where $K = \left (k_{ij}\right)\in \mathbb{C}^{m\times m}, \ k_{ij} = \dfrac{1}{1-(\alpha_{i}\alpha_{j})^{2}}, \ (i, j = 1, 2, \cdots, m).$

Proof. For $X = (x_{ij})\in \mathbb{C}^{m\times m}, J = (J_{ij})\in \mathbb{C}^{m\times m},$ $(2.3)$ can be equivalently written as

By $(2.5)$, we can get

Then the equation of $(2.4)$ follows from $(2.6).$

Assume that the canonical correlation decomposition (CCD) ^[20] of the matrix pair $[A^{\ast}, B^{\ast}]$ is

where $E_{A} \in \mathbb{C}^{n \times n}$ and $E_{B} \in \mathbb{C}^{p \times p}$ are nonsingular matrices, and

with

$q = \mbox{rank}(A)+\mbox{rank}(B)-\mbox{rank}(A^{\ast}, B^{\ast}), \ g = \mbox{rank}(A) = q+s+t, \ h = \mbox{rank}(B),$ and $Q = [Q_{1}, Q_{2}, Q_{3}, Q_{4}, Q_{5}, Q_{6}]\in \mathbb{UC} ^{m \times m}$ with the partition of $Q$ being compatible with those of $\Sigma_{A}$ and $\Sigma_{B}.$

According to (2.7), (1.2) can be equivalently written as

partition the matrices $E_{A}^{-1}X(E_{B}^{-1})^{\ast}$ and $Q^{\ast}DQ$ into the following forms:

where $u = h-q-s, v = m-h-s-t.$ Inserting (2.9) and (2.10) into (2.8), we have

Apparently, $\Phi_{1} = \min$ if and only if

By (2.11), we have

where $N \in\mathbb{C}^{q\times q}$ is some skew-Hermitian matrix. According to (2.12) and Lemma 2.1, we have

By (2.13) and Lemmas 2.2 and 2.3, we have

where $K = \left (k_{ij}\right)\in\mathbb{C}^{s\times s}, k_{ij} = \dfrac{1}{1-\left (\alpha_{i}\alpha_{j}\right)^{2}}, \ (i, j = 1, 2, \cdots, s),$ $J = C_{A}D_{22}+S_{A}D_{25}^{\ast}.$ Inserting (2.14)–(2.16) into $(2.9)$, we can get the following result.

Theorem 2.1. Suppose that $A \in\mathbb{C}^{n\times m}, B \in\mathbb{C}^{p\times m}, D \in\mathbb{C}^{m\times m}$ with $D = D^{\ast}.$ Let the canonical correlation decomposition of the matrix pair $[A^{\ast}, B^{\ast}]$ be given by $(2.7),$ the partition of the matrices $E_{A}^{-1}X(E_{B}^{-1})^{\ast}, Q^{\ast}DQ$ be given by $(2.9)$ and $(2.10),$ respectively. Then, the general solution of Problem I can be expressed as

where $X_{22}$ is given by $(2.16)$, $X_{i4}, X_{4j}, (i = 1, 2, 3, 4;j = 1, 2, 3)$ are arbitrary matrices and $N \in\mathbb{C}^{q\times q}$ is some skew-Hermitian matrix.

Clearly, (1.1) with $D = D^{\ast}$ is sovable if and only if $\Phi_{1} = \min = 0,$ that is,

$D_{i4} = 0, D_{4j} = 0, \ (i = 1, 2, 3, 4; j = 5, 6), D_{33} = 0, D_{55} = 0, D_{56} = 0, D_{66} = 0.$ By $(2.17),$ we obtain

where $N\in\mathbb{C}^{q\times q}$ is some skew-Hermitian matrix. Inserting (2.18) and (2.19) into (2.9), we have the following result.

Theorem 2.2. Suppose that $A \in\mathbb{C}^{n\times m}, B \in\mathbb{C}^{p\times m}, D \in\mathbb{C}^{m\times m}$ with $D = D^{\ast}.$ Let the canonical correlation decomposition(CCD) of the matrix pair $[A^{\ast}, B^{\ast}]$ be given by $(2.7),$ the partition of the matrices $E_{A}^{-1}X(E_{B}^{-1})^{\ast}, Q^{\ast}DQ$ be given by $(2.9)$ and $(2.10),$ respectively. Then $(1.1)$ has a solution if and only if

In this case, the general solution of $(1.1)$ can be expressed as

where $X_{i4}, X_{4j}, (i = 1, 2, 3, 4; j = 1, 2, 3)$ are arbitrary matrices and $N \in\mathbb{C}^{q\times q}$ is some skew-Hermitian matrix.

Remark 2.1. Note that $(1.1)$ has a unique solution $X$ if and only if

which is equivalent to

In this case, the unique solution of $(1.1)$ can be expressed as

Based on Theorem 2.2, we can formulate the following algorithm $2.1$ to solve (1.1).

Algorithm 2.1.

$1)$ Input matrices $A, B$ and $D$ with $D = D^\ast$.

$2)$ Compute the canonical correlation decomposition of the matrix pair $[A^{\ast}, B^{\ast}]$ by $(2.7)$.

$3)$ Compute $D_{11}, D_{12}, D_{13}, D_{15}, D_{16}, D_{22}, D_{23}, D_{25}, D_{26}, D_{35}$ and $D_{36}$ by $(2.10)$, respectively.

$4)$ If the conditions $(2.20)$ are satisfied, go to $5)$; otherwise, Problem I has no solution, and stop.

$5)$ Randomly choose skew-Hermitian matrix $N\in {\mathbb{C}}^{q \times q}$ and compute $X_{11}$ by $(2.18)$.

$6)$ Compute $X_{12}, X_{13}, X_{21}, X_{22}, X_{23}, X_{31}, X_{32}, X_{33}$ by $(2.19)$, respectively.

$7)$ Randomly choose $X_{i4}, X_{4j}, (i = 1, 2, 3, 4; j = 1, 2, 3)$ and compute $X$ by $(2.21)$.

Example 2.1. Let $m = 10, n = 7, p = 6$. Suppose that the matrices $A, B$ and $D$ are given by

It is easy to verify the solvability conditions are satisfied:

According to Algorithm $2.1$ above, if choose $N = 0, X_{i4} = 0, X_{4j} = 0, (i = 1, 2, 3, 4; j = 1, 2, 3),$ then we can obtain a feasible solution $X$ of $(1.1)$ as follows:

with the corresponding residual estimated by

3.   Solution to Problem II

Suppose that the CCD of the matrix pair $[A^{\ast}, B^{\ast}]$ is of the form given by (2.7). For any $X \in\mathbb{C}^{n\times p}$, we define a weighted norm as follows ^[21,22,23] :

Write

Therefore, for any $X \in \mathcal{S}_{E}$, (1.3) can be written as

Obviously, $\left\|F-X\right\|_{W} = \min$, if and only if

where $M = F_{11}-\dfrac{1}{2}D_{11}$. Note that $N \in\mathbb{C}^{q\times q}$ is some skew-Hermitian matrix, which implies that the relation of (3.3) is equivalent to

therefore, we have

where $M = F_{11}-\dfrac{1}{2}D_{11}$. Apparently, by (3.4) and (3.5), we have

Summing up the discussions above, we have the following result.

Theorem 3.1. Given $F \in\mathbb{C}^{n\times p}$ and partition $E_{A}^{-1}F(E_{B}^{-1})^{\ast}$ as $(3.2)$. Let $M = F_{11}-\dfrac{1}{2}D_{11}$, then the unique solution of Problem II can be expressed as

Based on Theorem 3.1, we can formulate the following algorithm $3.1$ to solve Problem II.

Algorithm 3.1.

$1)$ Input matrices $A, B, F$ and $D$ with $D = D^\ast$.

$2)$ Compute the canonical correlation decomposition of the matrix pair $[A^{\ast}, B^{\ast}]$ by $(2.7)$.

$3)$ Compute $D_{11}, D_{12}, D_{13}, D_{15}, D_{16}, D_{22}, D_{23}, D_{25}, D_{26}, D_{35}$ and $D_{36}$ by $(2.10)$, respectively.

$4)$ If the conditions $(2.20)$ are satisfied, go to $5)$; otherwise, Problem II has no solution, and stop.

$5)$ Compute $F_{11}, F_{i4}, F_{4j} \ (i = 1, 2, 3, 4; j = 1, 2, 3)$ by $(3.2)$.

$6)$ Set $M = F_{11}-\dfrac{1}{2}D_{11}$ and compute $X_{11}$ by $(3.6)$.

$7)$ Compute $X_{i4}$ and $X_{4j} \; (i = 1, 2, 3, 4; j = 1, 2, 3)$ by $(3.7)$ and compute $\hat{X}$ by $(3.8)$.

Example 3.1. Let $m = 10, n = 7, p = 6$. Suppose that the matrices $A, B, D$ and $F$ are given by

and the matrices $A, B, D$ are the same as those of Example $2.1$. It is easy to verify the solvability conditions are satisfied by Example $2.1$. According to Algorithm $3.1$ above, we can obtain the unique solution $\hat{X}$ of Problem II as follows:

with the corresponding residual estimated by

4.   Conclusions

In this paper, we have obtained the expression of the least-squares solutions to Problem I and the unique solution $\hat{X}$ of Problem II by using the CCD of the matrix pair $[A^{\ast}, B^{\ast}]$. Two numerical examples verify the correctness of our results.

Conflict of interest

All authors declare that there is no conflict of interest in this paper.