Constrained Solutions of a System of Matrix Equations

Abstract

We derive the necessary and sufficient conditions of and the expressions for the orthogonal solutions, the symmetric orthogonal solutions, and the skew-symmetric orthogonal solutions of the system of matrix equations 𝐴𝑋=𝐵 and 𝑋𝐶=𝐷, respectively. When the matrix equations are not consistent, the least squares symmetric orthogonal solutions and the least squares skew-symmetric orthogonal solutions are respectively given. As an auxiliary, an algorithm is provided to compute the least squares symmetric orthogonal solutions, and meanwhile an example is presented to show that it is reasonable.

1. Introduction

Throughout this paper, the following notations will be used. ℝ^𝑚×𝑛, 𝑂ℝ^𝑛×𝑛, 𝑆ℝ^𝑛×𝑛, and 𝐴𝑆ℝ^𝑛×𝑛 denote the set of all 𝑚×𝑛 real matrices, the set of all 𝑛×𝑛 orthogonal matrices, the set of all 𝑛×𝑛 symmetric matrices, and the set of all 𝑛×𝑛 skew-symmetric matrices, respectively. 𝐼_𝑛 is the identity matrix of order 𝑛. (⋅)^𝑇 and tr(⋅) represent the transpose and the trace of the real matrix, respectively. ‖⋅‖ stands for the Frobenius norm induced by the inner product. The following two definitions will also be used.

Definition 1.1 (see [1]). A real matrix 𝑋∈ℝ^𝑛×𝑛 is said to be a symmetric orthogonal matrix if 𝑋^𝑇=𝑋 and 𝑋^𝑇𝑋=𝐼_𝑛.

Definition 1.2 (see [2]). A real matrix 𝑋∈ℝ^2𝑚×2𝑚 is called a skew-symmetric orthogonal matrix if 𝑋^𝑇=−𝑋 and 𝑋^𝑇𝑋=𝐼_𝑛.

The set of all 𝑛×𝑛 symmetric orthogonal matrices and the set of all 2𝑚×2𝑚 skew-symmetric orthogonal matrices are, respectively, denoted by 𝑆𝑂ℝ^𝑛×𝑛 and 𝑆𝑆𝑂ℝ^2𝑚×2𝑚. Since the linear matrix equation(s) and its optimal approximation problem have great applications in structural design, biology, control theory, and linear optimal control, and so forth, see, for example, [3–5], there has been much attention paid to the linear matrix equation(s). The well-known system of matrix equations 𝐴𝑋=𝐵,𝑋𝐶=𝐷, (1.1) as one kind of linear matrix equations, has been investigated by many authors, and a series of important and useful results have been obtained. For instance, the system (1.1) with unknown matrix 𝑋 being bisymmetric, centrosymmetric, bisymmetric nonnegative definite, Hermitian and nonnegative definite, and (𝑃,𝑄)-(skew) symmetric has been, respectively, investigated by Wang et al. [6, 7], Khatri and Mitra [8], and Zhang and Wang [9]. Of course, if the solvability conditions of system (1.1) are not satisfied, we may consider its least squares solution. For example, Li et al. [10] presented the least squares mirrorsymmetric solution. Yuan [11] got the least-squares solution. Some results concerning the system (1.1) can also be found in [12–18].

Symmetric orthogonal matrices and skew-symmetric orthogonal matrices play important roles in numerical analysis and numerical solutions of partial differential equations. Papers [1, 2], respectively, derived the symmetric orthogonal solution 𝑋 of the matrix equation 𝑋𝐶=𝐷 and the skew-symmetric orthogonal solution 𝑋 of the matrix equation 𝐴𝑋=𝐵. Motivated by the work mentioned above, we in this paper will, respectively, study the orthogonal solutions, symmetric orthogonal solutions, and skew-symmetric orthogonal solutions of the system (1.1). Furthermore, if the solvability conditions are not satisfied, the least squares skew-symmetric orthogonal solutions and the least squares symmetric orthogonal solutions of the system (1.1) will be also given.

The remainder of this paper is arranged as follows. In Section 2, some lemmas are provided to give the main results of this paper. In Sections 3, 4, and 5, the necessary and sufficient conditions of and the expression for the orthogonal, the symmetric orthogonal, and the skew-symmetric orthogonal solutions of the system (1.1) are, respectively, obtained. In Section 6, the least squares skew-symmetric orthogonal solutions and the least squares symmetric orthogonal solutions of the system (1.1) are presented, respectively. In addition, an algorithm is provided to compute the least squares symmetric orthogonal solutions, and meanwhile an example is presented to show that it is reasonable. Finally, in Section 7, some concluding remarks are given.

2. Preliminaries

In this section, we will recall some lemmas and the special 𝐶-𝑆 decomposition which will be used to get the main results of this paper.

Lemma 2.1 (see [1, Lemmas 1 and 2]). Given 𝐶∈ℝ^2𝑚×𝑛,  𝐷∈ℝ^2𝑚×𝑛. The matrix equation 𝑌𝐶=𝐷 has a solution 𝑌∈𝑂ℝ^2𝑚×2𝑚 if and only if 𝐷^𝑇𝐷=𝐶^𝑇𝐶. Let the singular value decompositions of 𝐶 and 𝐷 be, respectively, 𝐶=𝑈𝑉Π000^𝑇,𝐷=𝑊𝑉Π000^𝑇, (2.1) where Π=diag𝜎₁,…,𝜎_𝑘(𝐶)(𝐷),𝑈>0,𝑘=rank=rank𝑈=₁𝑈₂∈𝑂ℝ^2𝑚×2𝑚,𝑈₁∈ℝ^2𝑚×𝑘,𝑊=𝑊₁𝑊₂∈𝑂ℝ^2𝑚×2𝑚,𝑊₁∈ℝ^2𝑚×𝑘,𝑉=𝑉₁𝑉₂∈𝑂ℝ^𝑛×𝑛,𝑉₁∈ℝ^𝑛×𝑘. (2.2) Then the orthogonal solutions of 𝑌𝐶=𝐷 can be described as 𝐼𝑌=𝑊^𝑘0𝑈0𝑃^𝑇, (2.3) where 𝑃∈𝑂ℝ^{(2𝑚−𝑘)×(2𝑚−𝑘)} is arbitrary.

Lemma 2.2 (see [2, Lemmas 1 and 2]). Given 𝐴∈ℝ^𝑛×𝑚,  𝐵∈ℝ^𝑛×𝑚. The matrix equation 𝐴𝑋=𝐵 has a solution 𝑋∈𝑂ℝ^𝑚×𝑚 if and only if 𝐴𝐴^𝑇=𝐵𝐵^𝑇. Let the singular value decompositions of 𝐴 and 𝐵 be, respectively, 𝐴=𝑈𝑉Σ000^𝑇,𝐵=𝑈𝑄Σ000^𝑇, (2.4) where Σ=diag𝛿₁,…,𝛿_𝑙>0,𝑙=rank(𝐴)=rank(𝐵),𝑈=𝑈₁𝑈₂∈𝑂ℝ^𝑛×𝑛,𝑈₁∈ℝ^𝑛×𝑙,𝑉=𝑉₁𝑉₂∈𝑂ℝ^𝑚×𝑚,𝑉₁∈ℝ^𝑚×𝑙,𝑄=𝑄₁𝑄₂∈𝑂ℝ^𝑚×𝑚,𝑄₁∈ℝ^𝑚×𝑙. (2.5) Then the orthogonal solutions of 𝐴𝑋=𝐵 can be described as 𝐼𝑋=𝑉^𝑙0𝑄0𝑊^𝑇, (2.6) where 𝑊∈𝑂ℝ^{(𝑚−𝑙)×(𝑚−𝑙)} is arbitrary.

Lemma 2.3 (see [2, Theorem 1]). If 𝑋𝑋=¹¹𝑋₁₂𝑋₂₁𝑋₂₂∈𝑂ℝ^2𝑚×2𝑚,𝑋₁₁∈𝐴𝑆ℝ^𝑘×𝑘, (2.7) then the 𝐶- 𝑆 decomposition of 𝑋 can be expressed as 𝐷₁00𝐷₂^𝑇𝑋₁₁𝑋₁₂𝑋₂₁𝑋₂₂𝐷₁00𝑅₂=Σ₁₁Σ₁₂Σ₂₁Σ₂₂, (2.8) where 𝐷₁∈𝑂ℝ^𝑘×𝑘,𝐷₂,𝑅₂∈𝑂ℝ^{(2𝑚−𝑘)×(2𝑚−𝑘)}, Σ₁₁=̃0𝐼00𝐶0000,Σ₁₂=,Σ0000𝑆000𝐼₂₁=0000𝑆000𝐼,Σ₂₂=𝐶𝐼000−^𝑇0,̃𝐼̃000𝐼=diag₁,…,𝐼̃_𝑟1,𝐼₁̃=⋯=𝐼̃_𝑟1=,𝐶01−10𝐶=diag₁,…,𝐶_𝑙1,𝐶_𝑖=0𝑐_𝑖−𝑐_𝑖0,𝑖=1,…,𝑙₁;𝑆=diag𝑆₁,…,𝑆_𝑙1,𝑆_𝑖=𝑠_𝑖00𝑠_𝑖,𝑆^𝑇_𝑖𝑆_𝑖+𝐶^𝑇_𝑖𝐶_𝑖=𝐼₂,𝑖=1,…,𝑙₁;2𝑟₁+2𝑙₁=rank𝑋₁₁,Σ₂₁=Σ^𝑇₁₂,𝑘−2𝑟₁=rank𝑋₂₁. (2.9)

Lemma 2.4 (see [1, Theorem 1]). If 𝐾𝐾=¹¹𝐾₁₂𝐾₂₁𝐾₂₂∈𝑂ℝ^𝑚×𝑚,𝐾₁₁∈𝑆ℝ^𝑙×𝑙, (2.10) then the 𝐶-𝑆 decomposition of 𝐾 can be described as 𝐷₁00𝐷₂^𝑇𝐾₁₁𝐾₁₂𝐾₂₁𝐾₂₂𝐷₁00𝑅₂=Π₁₁Π₁₂Π₂₁Π₂₂, (2.11) where 𝐷₁∈𝑂ℝ^𝑙×𝑙,  𝐷₂,  𝑅₂∈𝑂ℝ^{(𝑚−𝑙)×(𝑚−𝑙)}, Π₁₁=̃0𝐼00𝐶0000,Π₁₂=,Π0000𝑆000𝐼₂₁=0000𝑆000𝐼,Π₂₂=,̃𝑖𝐼000−𝐶0000𝐼=diag₁,…,𝑖_𝑟,𝑖_𝑗𝑠=±1,𝑗=1,…,𝑟;𝑆=diag_1′,…,𝑠_𝑙′,𝐶=diag𝑐_1′,…,𝑐_𝑙′,𝑠²_𝑖′=1−𝑐²_𝑖′,𝑖=1^′,…,𝑙^′;𝑟+𝑙^′=rank𝐾₁₁,Π₂₁=Π^𝑇₁₂. (2.12)

Remarks 2.5. In order to know the 𝐶- 𝑆 decomposition of an orthogonal matrix with a 𝑘×𝑘 leading (skew-) symmetric submatrix for details, one can deeply study the proof of Theorem  1 in [1] and [2].

Lemma 2.6. Given 𝐴∈ℝ^𝑛×𝑚,  𝐵∈ℝ^𝑛×𝑚. Then the matrix equation 𝐴𝑋=𝐵 has a solution 𝑋∈𝑆𝑂ℝ^𝑚×𝑚 if and only if 𝐴𝐴^𝑇=𝐵𝐵^𝑇 and 𝐴𝐵^𝑇=𝐵𝐴^𝑇. When these conditions are satisfied, the general symmetric orthogonal solutions can be expressed as 𝑉𝐼𝑋=^2𝑙−𝑟0𝑄0𝐺^𝑇, (2.13) where 𝑄=𝐽𝑄diag𝐼,𝐷₂∈𝑂ℝ^𝑚×𝑚,𝑉=𝐽𝑉diag𝐼,𝑅₂∈𝑂ℝ^𝑚×𝑚,𝐽=,𝐼0000𝐼0𝐼0 (2.14) and 𝐺∈𝑆𝑂ℝ^{(𝑚−2𝑙+𝑟)×(𝑚−2𝑙+𝑟)} is arbitrary.

Proof. The Necessity. Assume 𝑋∈𝑆𝑂ℝ^𝑚×𝑚 is a solution of the matrix equation 𝐴𝑋=𝐵, then we have 𝐵𝐵^𝑇=𝐴𝑋𝑋^𝑇𝐴^𝑇=𝐴𝐴^𝑇,𝐵𝐴^𝑇=𝐴𝑋𝐴^𝑇=𝐴𝑋^𝑇𝐴^𝑇=𝐴𝐵^𝑇. (2.15)
The Sufficiency. Since the equality 𝐴𝐴^𝑇=𝐵𝐵^𝑇 holds, then by Lemma 2.2, the singular value decompositions of 𝐴 and 𝐵 can be, respectively, expressed as (2.4). Moreover, the condition 𝐴𝐵^𝑇=𝐵𝐴^𝑇 means 𝑈𝑉Σ000^𝑇𝑄𝑈Σ000^𝑇=𝑈𝑄Σ000^𝑇𝑉𝑈Σ000^𝑇, (2.16) which can be written as 𝑉^𝑇₁𝑄₁=𝑄^𝑇₁𝑉₁. (2.17) From Lemma 2.2, the orthogonal solutions of the matrix equation 𝐴𝑋=𝐵 can be described as (2.6). Now we aim to find that 𝑋 in (2.6) is also symmetric. Suppose that 𝑋 is symmetric, then we have 𝐼_𝑙00𝑊^𝑇𝑉^𝑇𝑄=𝑄^𝑇𝑉𝐼_𝑙00𝑊, (2.18) together with the partitions of the matrices 𝑄 and 𝑉 in Lemma 2.2, we get 𝑉^𝑇₁𝑄₁=𝑄^𝑇₁𝑉₁,𝑄^𝑇₁𝑉₂𝑊=𝑉^𝑇₁𝑄₂,𝑄^𝑇₂𝑉₂𝑊=𝑊^𝑇𝑉^𝑇₂𝑄₂. (2.19)(2.20)(2.21) By (2.17), we can get (2.19). Now we aim to find the orthogonal solutions of the system of matrix equations (2.20) and (2.21). Firstly, we obtain from (2.20) that 𝑄^𝑇₁𝑉₂(𝑄^𝑇₁𝑉₂)^𝑇=𝑉^𝑇₁𝑄₂(𝑉^𝑇₁𝑄₂)^𝑇, then by Lemma 2.2, (2.20) has an orthogonal solution 𝑊. By (2.17), the 𝑙×𝑙 leading principal submatrix of the orthogonal matrix 𝑉^𝑇𝑄 is symmetric. Then we have, from Lemma 2.4, 𝑉^𝑇₁𝑄₂=𝐷₁Π₁₂𝑅^𝑇₂,𝑄^𝑇₁𝑉₂=𝐷₁Π₁₂𝐷^𝑇₂,𝑉^𝑇₂𝑄₂=𝐷₂Π₂₂𝑅^𝑇₂. (2.22)(2.23)(2.24) From (2.20), (2.22), and (2.23), the orthogonal solution 𝑊 of (2.20) is 𝑊=𝐷₂𝑅𝐺000𝐼000𝐼^𝑇2, (2.25) where 𝐺∈𝑂ℝ^{(𝑚−2𝑙+𝑟)×(𝑚−2𝑙+𝑟)} is arbitrary. Combining (2.21), (2.24), and (2.25) yields 𝐺^𝑇=𝐺,thatis,𝐺 is a symmetric orthogonal matrix. Denote 𝑉=𝑉diag𝐼,𝐷₂,𝑄=𝑄diag𝐼,𝑅₂, (2.26) then the symmetric orthogonal solutions of the matrix equation 𝐴𝑋=𝐵 can be expressed as 𝑋=𝑉𝑄𝐼000𝐺000𝐼^𝑇. (2.27) Let the partition matrix 𝐽 be 𝐽=𝐼0000𝐼0𝐼0, (2.28) compatible with the block matrix 𝐼000𝐺000𝐼. (2.29) Put 𝑉=𝐽𝑉,𝑄=𝐽𝑄, (2.30) then the symmetric orthogonal solutions of the matrix equation 𝐴𝑋=𝐵 can be described as (2.13).

Setting 𝐴=𝐶^𝑇,  𝐵=𝐷^𝑇, and 𝑋=𝑌^𝑇 in [2, Theorem 2], and then by Lemmas 2.1 and 2.3, we can have the following result.

Lemma 2.7. Given 𝐶∈ℝ^2𝑚×𝑛,  𝐷∈ℝ^2𝑚×𝑛. Then the equation has a solution 𝑌∈𝑆𝑆𝑂ℝ^2𝑚×2𝑚 if and only if 𝐷^𝑇𝐷=𝐶^𝑇𝐶 and 𝐷^𝑇𝐶=−𝐶^𝑇𝐷. When these conditions are satisfied, the skew-symmetric orthogonal solutions of the matrix equation 𝑌𝐶=𝐷 can be described as 𝑌=𝑊𝑈𝐼00𝐻^𝑇, (2.31) where 𝑊=𝐽^′𝑊diag𝐼,−𝐷₂∈𝑂ℝ^2𝑚×2𝑚,𝑈=𝐽^′𝑈diag𝐼,𝑅₂∈𝑂ℝ^2𝑚×2𝑚,𝐽^′=,𝐼0000𝐼0𝐼0 (2.32) and 𝐻∈𝑆𝑆𝑂ℝ^2𝑟×2𝑟 is arbitrary.

3. The Orthogonal Solutions of the System (1.1)

The following theorems give the orthogonal solutions of the system (1.1).

Theorem 3.1. Given 𝐴,  𝐵∈ℝ^𝑛×𝑚 and 𝐶,  𝐷∈ℝ^𝑚×𝑛, suppose the singular value decompositions of 𝐴 and 𝐵 are, respectively, as (2.4). Denote 𝑄^𝑇𝐶𝐶=¹𝐶₂,𝑉^𝑇𝐷𝐷=¹𝐷₂, (3.1) where 𝐶₁, 𝐷₁∈ℝ^𝑙×𝑛, and 𝐶₂,𝐷₂∈ℝ^{(𝑚−𝑙)×𝑛}. Let the singular value decompositions of 𝐶₂ and 𝐷₂ be, respectively, 𝐶₂=𝑈𝑉Π000^𝑇,𝐷₂=𝑊𝑉Π000^𝑇, (3.2) where 𝑈,  𝑊∈𝑂ℝ^{(𝑚−𝑙)×(𝑚−𝑙)},  𝑉∈𝑂ℝ^𝑛×𝑛,  Π∈ℝ^𝑘×𝑘 is diagonal, whose diagonal elements are nonzero singular values of 𝐶₂ or 𝐷₂. Then the system (1.1) has orthogonal solutions if and only if 𝐴𝐴^𝑇=𝐵𝐵^𝑇,𝐶₁=𝐷₁,𝐷^𝑇₂𝐷₂=𝐶^𝑇₂𝐶₂. (3.3) In which case, the orthogonal solutions can be expressed as 𝑉𝐼𝑋=^𝑘+𝑙00𝐺^′𝑄^𝑇, (3.4) where 𝐼𝑉=𝑉^𝑙00𝑊∈𝑂ℝ^𝑚×𝑚,𝐼𝑄=𝑄^𝑙00𝑈∈𝑂ℝ^𝑚×𝑚, (3.5) and 𝐺^′∈𝑂ℝ^{(𝑚−𝑘−𝑙)×(𝑚−𝑘−𝑙)} is arbitrary.

Proof. Let the singular value decompositions of 𝐴 and 𝐵 be, respectively, as (2.4). Since the matrix equation 𝐴𝑋=𝐵 has orthogonal solutions if and only if 𝐴𝐴^𝑇=𝐵𝐵^𝑇, (3.6) then by Lemma 2.2, its orthogonal solutions can be expressed as (2.6). Substituting (2.6) and (3.1) into the matrix equation 𝑋𝐶=𝐷, we have 𝐶₁=𝐷₁ and 𝑊𝐶₂=𝐷₂. By Lemma 2.1, the matrix equation 𝑊𝐶₂=𝐷₂ has orthogonal solution 𝑊 if and only if 𝐷^𝑇₂𝐷₂=𝐶^𝑇₂𝐶₂. (3.7) Let the singular value decompositions of 𝐶₂ and 𝐷₂ be, respectively, 𝐶₂=𝑈𝑉Π000^𝑇,𝐷₂=𝑊𝑉Π000^𝑇, (3.8) where 𝑈,  𝑊∈𝑂ℝ^{(𝑚−𝑙)×(𝑚−𝑙)},  𝑉∈𝑂ℝ^𝑛×𝑛,  Π∈ℝ^𝑘×𝑘 is diagonal, whose diagonal elements are nonzero singular values of 𝐶₂ or 𝐷₂. Then the orthogonal solutions can be described as 𝑊𝐼𝑊=^𝑘00𝐺^′𝑈^𝑇, (3.9) where 𝐺^′∈𝑂ℝ^{(𝑚−𝑘−𝑙)×(𝑚−𝑘−𝑙)} is arbitrary. Therefore, the common orthogonal solutions of the system (1.1) can be expressed as 𝐼𝑋=𝑉^𝑙0𝑄0𝑊^𝑇𝐼=𝑉^𝑙00𝑊𝐼_𝑙000𝐼_𝑘000𝐺^′𝐼_𝑙00𝑈^𝑇𝑄^𝑇=𝑉𝐼_𝑘+𝑙00𝐺^′𝑄^𝑇, (3.10) where 𝐼𝑉=𝑉^𝑙00𝑊∈𝑂ℝ^𝑚×𝑚,𝐼𝑄=𝑄^𝑙00𝑈∈𝑂ℝ^𝑚×𝑚, (3.11) and 𝐺^′∈𝑂ℝ^{(𝑚−𝑘−𝑙)×(𝑚−𝑘−𝑙)} is arbitrary.

The following theorem can be shown similarly.

Theorem 3.2. Given 𝐴,  𝐵∈ℝ^𝑛×𝑚 and 𝐶,  𝐷∈ℝ^𝑚×𝑛, let the singular value decompositions of 𝐶 and 𝐷 be, respectively, as (2.1). Partition 𝐴𝑊=𝐴₁𝐴₂,𝐵𝑈=𝐵₁𝐵₂, (3.12) where 𝐴₁,𝐵₁∈ℝ^𝑛×𝑘,  𝐴₂,𝐵₂∈ℝ^{𝑛×(𝑚−𝑘)}. Assume the singular value decompositions of 𝐴₂ and 𝐵₂ are, respectively, 𝐴₂=𝑈𝑉Σ000^𝑇,𝐵₂=𝑈𝑄Σ000^𝑇, (3.13) where 𝑉,  𝑄∈𝑂ℝ^{(𝑚−𝑘)×(𝑚−𝑘)},  𝑈∈𝑂ℝ^𝑛×𝑛,  Σ∈ℝ^{𝑙′×𝑙′}is diagonal, whose diagonal elements are nonzero singular values of 𝐴₂ or 𝐵₂. Then the system (1.1) has orthogonal solutions if and only if 𝐷^𝑇𝐷=𝐶^𝑇𝐶,𝐴₁=𝐵₁,𝐴₂𝐴^𝑇₂=𝐵₂𝐵^𝑇₂. (3.14) In which case, the orthogonal solutions can be expressed as 𝑊𝐼𝑋=^𝑘+𝑙′00𝐻^′𝑈^𝑇, (3.15) where 𝐼𝑊=𝑊^𝑘00𝑊∈𝑂ℝ^𝑚×𝑚,𝐼𝑈=𝑈^𝑘00𝑈∈𝑂ℝ^𝑚×𝑚, (3.16) and 𝐻^′∈𝑂ℝ^{(𝑚−𝑘−𝑙′)×(𝑚−𝑘−𝑙′)} is arbitrary.

4. The Symmetric Orthogonal Solutions of the System (1.1)

We now present the symmetric orthogonal solutions of the system (1.1).

Theorem 4.1. Given 𝐴,𝐵∈ℝ^𝑛×𝑚,  𝐶,𝐷∈ℝ^𝑚×𝑛. Let the symmetric orthogonal solutions of the matrix equation 𝐴𝑋=𝐵 be described as in Lemma 2.6. Partition 𝑄^𝑇𝐶𝐶=^′1𝐶^′₂,𝑉^𝑇𝐷𝐷=^′1𝐷^′₂, (4.1) where 𝐶^′₁,𝐷^′₁∈ℝ^{(2𝑙−𝑟)×𝑛},  𝐶^′₂,𝐷^′₂∈ℝ^{(𝑚−2𝑙+𝑟)×𝑛}. Then the system (1.1) has symmetric orthogonal solutions if and only if 𝐴𝐴^𝑇=𝐵𝐵^𝑇,𝐴𝐵^𝑇=𝐵𝐴^𝑇,𝐶^′₁=𝐷^′₁,𝐷₂^′𝑇𝐷^′₂=𝐶₂^′𝑇𝐶^′₂,𝐷₂^′𝑇𝐶^′₂=𝐶₂^′𝑇𝐷^′₂. (4.2) In which case, the solutions can be expressed as 𝑋=𝑉𝐼00𝐺^′′𝑄^𝑇, (4.3) where 𝐼𝑉=𝑉^2𝑙−𝑟00𝑊∈𝑂ℝ^𝑚×𝑚,𝐼𝑄=𝑄^2𝑙−𝑟00𝑈∈𝑂ℝ^𝑚×𝑚, (4.4) and 𝐺^′′∈𝑆𝑂ℝ^{(𝑚−2𝑙+𝑟−2𝑙′+𝑟′)×(𝑚−2𝑙+𝑟−2𝑙′+𝑟′)} is 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦.

Proof. From Lemma 2.6, we obtain that the matrix equation 𝐴𝑋=𝐵 has symmetric orthogonal solutions if and only if 𝐴𝐴^𝑇=𝐵𝐵^𝑇 and 𝐴𝐵^𝑇=𝐵𝐴^𝑇. When these conditions are satisfied, the general symmetric orthogonal solutions can be expressed as 𝑉𝐼𝑋=^2𝑙−𝑟0𝑄0𝐺^𝑇, (4.5) where 𝐺∈𝑆𝑂ℝ^{(𝑚−2𝑙+𝑟)×(𝑚−2𝑙+𝑟)} is arbitrary, 𝑄∈𝑂ℝ^𝑚×𝑚,  𝑉∈𝑂ℝ^𝑚×𝑚. Inserting (4.1) and (4.5) into the matrix equation 𝑋𝐶=𝐷, we get 𝐶^′₁=𝐷^′₁ and 𝐺𝐶^′₂=𝐷^′₂. By [1, Theorem  2], the matrix equation 𝐺𝐶^′₂=𝐷^′₂ has a symmetric orthogonal solution if and only if 𝐷₂^′𝑇𝐷^′₂=𝐶₂^′𝑇𝐶^′₂,𝐷₂^′𝑇𝐶^′₂=𝐶₂^′𝑇𝐷^′₂. (4.6) In which case, the solutions can be described as 𝑊𝐼𝐺=^{2𝑙′−𝑟′}00𝐺^′′𝑈^𝑇, (4.7) where 𝐺^′′∈𝑆𝑂ℝ^{(𝑚−2𝑙+𝑟−2𝑙′+𝑟′)×(𝑚−2𝑙+𝑟−2𝑙′+𝑟′)} is arbitrary, 𝑊∈𝑂ℝ^{(𝑚−2𝑙+𝑟)×(𝑚−2𝑙+𝑟)}, and 𝑈∈𝑂ℝ^{(𝑚−2𝑙+𝑟)×(𝑚−2𝑙+𝑟)}. Hence the system (1.1) has symmetric orthogonal solutions if and only if all equalities in (4.2) hold. In which case, the solutions can be expressed as 𝑉𝐼𝑋=^2𝑙−𝑟0𝑄0𝐺^𝑇=𝑉𝐼_2𝑙−𝑟00𝑊𝐼_2𝑙−𝑟00𝐼00𝐺^′′𝐼_2𝑙−𝑟00𝑈^𝑇𝑄^𝑇, (4.8) that is, the expression in (4.3).

The following theorem can also be obtained by the method used in the proof of Theorem 4.1.

Theorem 4.2. Given 𝐴,𝐵∈ℝ^𝑛×𝑚,  𝐶,𝐷∈ℝ^𝑚×𝑛. Let the symmetric orthogonal solutions of the matrix equation 𝑋𝐶=𝐷 be described as 𝑀𝐼𝑋=^2𝑘−𝑟0𝑁0𝐺^𝑇, (4.9) where 𝑀,  𝑁∈𝑂ℝ^𝑚×𝑚,  𝐺∈𝑆𝑂ℝ^{(𝑚−2𝑘+𝑟)×(𝑚−2𝑘+𝑟)}. Partition 𝐴𝑀=𝑀₁𝑀₂,𝐵𝑁=𝑁₁𝑁₂,𝑀₁,𝑁₁∈ℝ^{𝑛×(2𝑘−𝑟)},𝑀₂,𝑁₂∈ℝ^{𝑛×(𝑚−2𝑘+𝑟)}. (4.10) Then the system (1.1) has symmetric orthogonal solutions if and only if 𝐷^𝑇𝐷=𝐶^𝑇𝐶,𝐷^𝑇𝐶=𝐶^𝑇𝐷,𝑀₁=𝑁₁,𝑀₂𝑀^𝑇₂=𝑁₂𝑁^𝑇₂,𝑀₂𝑁^𝑇₂=𝑁₂𝑀^𝑇₂. (4.11) In which case, the solutions can be expressed as 𝑋=𝑀𝐼00𝐻^′′𝑁^𝑇, (4.12) where 𝑀𝐼𝑀=^2𝑘−𝑟00𝑊₁∈𝑂ℝ^𝑚×𝑚,𝑁𝐼𝑁=^2𝑘−𝑟00𝑈₁∈𝑂ℝ^𝑚×𝑚, (4.13) and 𝐻^′′∈𝑆𝑂ℝ^{(𝑚−2𝑘+𝑟−2𝑘′+𝑟′)×(𝑚−2𝑘+𝑟−2𝑘′+𝑟′)} is arbitrary.

5. The Skew-Symmetric Orthogonal Solutions of the System (1.1)

In this section, we show the skew-symmetric orthogonal solutions of the system (1.1).

Theorem 5.1. Given 𝐴,𝐵∈ℝ^𝑛×2𝑚,  𝐶,𝐷∈ℝ^2𝑚×𝑛. Suppose the matrix equation 𝐴𝑋=𝐵 has skew-symmetric orthogonal solutions with the form 𝑋=𝑉₁𝑄𝐼00𝐻^𝑇1, (5.1) where 𝐻∈𝑆𝑆𝑂ℝ^2𝑟×2𝑟 is arbitrary, 𝑉₁,  𝑄₁∈𝑂ℝ^2𝑚×2𝑚. Partition 𝑄^𝑇₁𝑄𝐶=¹𝑄₂,𝑉^𝑇₁𝑉𝐷=¹𝑉₂, (5.2) where 𝑄₁,𝑉₁∈ℝ^{(2𝑚−2𝑟)×𝑛},  𝑄₂,𝑉₂∈ℝ^2𝑟×𝑛. Then the system (1.1) has skew-symmetric orthogonal solutions if and only if 𝐴𝐴^𝑇=𝐵𝐵^𝑇,𝐴𝐵^𝑇=−𝐵𝐴^𝑇,𝑄₁=𝑉₁,𝑄^𝑇₂𝑄₂=𝑉^𝑇₂𝑉₂,𝑄^𝑇₂𝑉₂=−𝑉^𝑇₂𝑄₂. (5.3) In which case, the solutions can be expressed as 𝑋=𝑉₁𝐼00𝐽^′𝑄^𝑇₁, (5.4) where 𝑉=𝑉₁𝐼_{2𝑚−2𝑟}00𝑊∈𝑂ℝ^2𝑚×2𝑚,𝑄=𝑄₁𝐼_{2𝑚−2𝑟}00𝑈∈𝑂ℝ^2𝑚×2𝑚, (5.5) and 𝐽^′∈𝑆𝑆𝑂ℝ^{2𝑘′×2𝑘′} is arbitrary.

Proof. By [2, Theorem 2], the matrix equation 𝐴𝑋=𝐵 has the skew-symmetric orthogonal solutions if and only if 𝐴𝐴^𝑇=𝐵𝐵^𝑇 and 𝐴𝐵^𝑇=−𝐵𝐴^𝑇. When these conditions are satisfied, the general skew-symmetric orthogonal solutions can be expressed as (5.1). Substituting (5.1) and (5.2) into the matrix equation 𝑋𝐶=𝐷, we get 𝑄₁=𝑉₁ and 𝐻𝑄₂=𝑉₂. From Lemma 2.7, equation 𝐻𝑄₂=𝑉₂ has a skew-symmetric orthogonal solution 𝐻 if and only if 𝑄^𝑇₂𝑄₂=𝑉^𝑇₂𝑉₂,𝑄^𝑇₂𝑉₂=−𝑉^𝑇₂𝑄₂. (5.6) When these conditions are satisfied, the solution can be described as 𝐻=𝑊𝐼00𝐽^′𝑈^𝑇, (5.7) where 𝐽^′∈𝑆𝑆𝑂ℝ^{2𝑘′×2𝑘′}is arbitrary, 𝑊,  𝑈∈𝑂ℝ^2𝑟×2𝑟. Inserting (5.7) into (5.1) yields that the system (1.1) has skew-symmetric orthogonal solutions if and only if all equalities in (5.3) hold. In which case, the solutions can be expressed as (5.4).

Similarly, the following theorem holds.

Theorem 5.2. Given 𝐴,𝐵∈ℝ^𝑛×2𝑚,  𝐶,𝐷∈ℝ^2𝑚×𝑛. Suppose the matrix equation 𝑋𝐶=𝐷 has skew-symmetric orthogonal solutions with the form 𝑋=𝑊𝑈𝐼00𝐾^𝑇, (5.8) where 𝐾∈𝑆𝑆𝑂ℝ^2𝑝×2𝑝 is arbitrary, 𝑊,  𝑈∈𝑂ℝ^2𝑚×2𝑚. Partition 𝐴𝑊=𝑊₁𝑊₂,𝐵𝑈=𝑈₁𝑈₂, (5.9) where 𝑊₁,𝑈₁∈ℝ^{𝑛×(2𝑚−2𝑝)},  𝑊₂,𝑈₂∈ℝ^𝑛×2𝑝. Then the system (1.1) has skew-symmetric orthogonal solutions if and only if 𝐷^𝑇𝐷=𝐶^𝑇𝐶,𝐷^𝑇𝐶=−𝐶^𝑇𝐷,𝑊₁=𝑈₁,𝑊₂𝑊^𝑇₂=𝑈₂𝑈^𝑇₂,𝑈₂𝑊^𝑇₂=−𝑊₂𝑈^𝑇₂. (5.10) In which case, the solutions can be expressed as 𝑋=𝑊₁𝐼00𝐽^′′𝑈^𝑇₁, (5.11) where 𝑊₁=𝑊𝐼_{2𝑚−2𝑝}00𝑊₁∈𝑂ℝ^2𝑚×2𝑚,𝑈₁=𝐼_{2𝑚−2𝑝}00𝑈^𝑇₁𝑈∈𝑂ℝ^2𝑚×2𝑚, (5.12) and 𝐽^′′∈𝑆𝑆𝑂ℝ^2𝑞×2𝑞 is arbitrary.

6. The Least Squares (Skew-) Symmetric Orthogonal Solutions of the System (1.1)

If the solvability conditions of a system of matrix equations are not satisfied, it is natural to consider its least squares solution. In this section, we get the least squares (skew-) symmetric orthogonal solutions of the system (1.1), that is, seek 𝑋∈𝑆𝑆𝑂ℝ^2𝑚×2𝑚(𝑆𝑂ℝ^𝑛×𝑛) such that min_{𝑋∈𝑆𝑆𝑂ℝ2𝑚×2𝑚(𝑆𝑂ℝ𝑛×𝑛)}‖𝐴𝑋−𝐵‖²+‖𝑋𝐶−𝐷‖². (6.1) With the help of the definition of the Frobenius norm and the properties of the skew-symmetric orthogonal matrix, we get that ‖𝐴𝑋−𝐵‖²+‖𝑋𝐶−𝐷‖²=‖𝐴‖²+‖𝐵‖²+‖𝐶‖²+‖𝐷‖²−2tr𝑋^𝑇𝐴^𝑇𝐵+𝐷𝐶^𝑇. (6.2) Let 𝐴^𝑇𝐵+𝐷𝐶^𝑇𝐾=𝐾,_𝑇−𝐾2=𝑇. (6.3) Then, it follows from the skew-symmetric matrix 𝑋 that tr𝑋^𝑇𝐴^𝑇𝐵+𝐷𝐶^𝑇=tr𝑋^𝑇𝐾()𝑋𝐾=tr−𝑋𝐾=tr^𝑇−𝐾2=tr(𝑋𝑇). (6.4) Therefore, (6.1) holds if and only if (6.4) reaches its maximum. Now, we pay our attention to find the maximum value of (6.4). Assume the eigenvalue decomposition of 𝑇 is 𝑇=𝐸𝐸Λ000^𝑇 (6.5) with Λ=diagΛ₁,…,Λ_𝑙,Λ_𝑖=0𝛼_𝑖−𝛼_𝑖0,𝛼_𝑖>0,𝑖=1,…,𝑙;2𝑙=rank(𝑇). (6.6) Denote 𝐸^𝑇𝑋𝑋𝐸=¹¹𝑋₁₂𝑋₂₁𝑋₂₂, (6.7) partitioned according to Λ000, (6.8) then (6.4) has the following form: tr(𝑋𝑇)=tr𝐸^𝑇𝑋𝐸𝑋Λ000=tr₁₁Λ. (6.9) Thus, by 𝑋₁₁=𝐼̃=diag𝐼₁̃,…,𝐼_𝑙̃, (6.10) where 𝐼_𝑖̃=0−110,𝑖=1,…,𝑙. (6.11) Equation (6.9) gets its maximum. Since 𝐸^𝑇𝑋𝐸 is skew-symmetric, it follows from ̃𝑋=𝐸𝐸𝐼00𝐺^𝑇, (6.12) where 𝐺∈𝑆𝑆𝑂ℝ^{(2𝑚−2𝑙)×(2𝑚−2𝑙)} is arbitrary, that (6.1) obtains its minimum. Hence we have the following theorem.

Theorem 6.1. Given 𝐴,𝐵∈ℝ^𝑛×2𝑚 and 𝐶,𝐷∈ℝ^2𝑚×𝑛, denote 𝐴^𝑇𝐵+𝐷𝐶^𝑇𝐾=𝐾,^𝑇−𝐾2=𝑇, (6.13) and let the spectral decomposition of 𝑇 be (6.5). Then the least squares skew-symmetric orthogonal solutions of the system (1.1) can be expressed as (6.12).

If 𝑋 in (6.1) is a symmetric orthogonal matrix, then by the definition of the Frobenius norm and the properties of the symmetric orthogonal matrix, (6.2) holds. Let 𝐴^𝑇𝐵+𝐷𝐶^𝑇𝐻=𝐻,^𝑇+𝐻2=𝑁. (6.14) Then we get that min_{𝑋∈𝑆𝑂ℝ𝑛×𝑛}‖𝐴𝑋−𝐵‖²+‖𝑋𝐶−𝐷‖²=min_{𝑋∈𝑆𝑂ℝ𝑛×𝑛}‖𝐴‖²+‖𝐵‖²+‖𝐶‖²+‖𝐷‖²−2tr(𝑋𝑁). (6.15) Thus (6.15) reaches its minimum if and only if tr(𝑋𝑁) obtains its maximum. Now, we focus on finding the maximum value of tr(𝑋𝑁). Let the spectral decomposition of the symmetric matrix 𝑁 be 𝑁=𝑀𝑀Σ000^𝑇, (6.16) where ΣΣ=⁺00Σ⁻,Σ⁺=diag𝜆₁,…,𝜆_𝑠,𝜆₁≥𝜆₂≥⋯≥𝜆_𝑠>0,Σ⁻=diag𝜆_𝑠+1,…,𝜆_𝑡,𝜆_𝑡≤𝜆_𝑡−1≤⋯≤𝜆_𝑠+1<0,𝑡=rank(𝑁). (6.17) Denote 𝑀^𝑇𝑋𝑀=𝑋₁₁𝑋₁₂𝑋₂₁𝑋₂₂ (6.18) being compatible with Σ000. (6.19) Then tr(𝑋𝑁)=tr𝑀^𝑇𝑋𝑀Σ000=tr𝑋₁₁𝑋₁₂𝑋₂₁𝑋₂₂Σ000=tr𝑋₁₁Σ. (6.20) Therefore, it follows from 𝑋₁₁=̂𝐼𝐼=^𝑠00−𝐼_𝑡−𝑠 (6.21) that (6.20) reaches its maximum. Since 𝑀^𝑇𝑋𝑀 is a symmetric orthogonal matrix, then when 𝑋 has the form ̂𝑋=𝑀𝑀𝐼00𝐿^𝑇, (6.22) where 𝐿∈𝑆𝑂ℝ^{(𝑛−𝑡)×(𝑛−𝑡)} is arbitrary, (6.15) gets its minimum. Thus we obtain the following theorem.

Theorem 6.2. Given 𝐴,𝐵∈ℝ^𝑛×𝑚,  𝐶,𝐷∈ℝ^𝑚×𝑛, denote 𝐴^𝑇𝐵+𝐷𝐶^𝑇𝐻=𝐻,^𝑇+𝐻2=𝑁, (6.23) and let the eigenvalue decomposition of 𝑁 be (6.16). Then the least squares symmetric orthogonal solutions of the system (1.1) can be described as (6.22).

Algorithm 6.3. Consider the following.𝑆𝑡𝑒𝑝1. Input 𝐴,𝐵∈ℝ^𝑛×𝑚 and 𝐶,𝐷∈ℝ^𝑚×𝑛.𝑆𝑡𝑒𝑝2. Compute 𝐴^𝑇𝐵+𝐷𝐶^𝑇𝐻=𝐻,𝑁=^𝑇+𝐻2. (6.24)𝑆𝑡𝑒𝑝3. Compute the spectral decomposition of 𝑁 with the form (6.16).𝑆𝑡𝑒𝑝4. Compute the least squares symmetric orthogonal solutions of (1.1) according to (6.22).

Example 6.4. Assume ,.𝐴=12.28.4−5.66.39.410.711.82.98.56.99.6−7.810.62.311.57.86.78.93.67.84.911.99.45.94.56.77.83.15.611.6,𝐵=8.59.43.67.86.34.72.73.67.99.45.67.83.76.78.69.83.42.9−4.36.25.77.45.49.52.93.9−5.26.37.84.6𝐶=5.46.43.75.69.73.64.27.8−6.37.86.73.5−4.62.92.8−2.77.210.83.73.81.93.98.25.611.28.97.89.47.95.6,𝐷=7.99.55.42.88.68.72.66.78.48.15.73.9−2.95.21.94.85.81.8−7.25.82.87.94.56.79.69.54.13.49.83.9 (6.25)

It can be verified that the given matrices 𝐴,𝐵,𝐶, and 𝐷 do not satisfy the solvability conditions in Theorem 4.1 or Theorem 4.2. So we intend to derive the least squares symmetric orthogonal solutions of the system (1.1). By Algorithm 6.3, we have the following results:(1)the least squares symmetric orthogonal solution 𝑋=0.012000.06621−0.249780.270470.91302−0.162180.066210.656010.22702−0.557690.245870.37715−0.249780.227020.806610.402540.04066−0.267840.27047−0.557690.402540.068530.237990.626450.913020.245870.040660.23799−0.12142−0.18135−0.162180.37715−0.267840.62645−0.1813540.57825, (6.26)(2)min_{𝑋∈𝑆𝑂ℝ𝑚×𝑚}‖𝐴𝑋−𝐵‖²+‖𝑋𝐶−𝐷‖²𝑋=1.98366,_𝑇𝑋−𝐼₆=2.84882×10⁻¹⁵,𝑋^𝑇−𝑋=0.00000. (6.27)

Remark 6.5. (1) There exists a unique symmetric orthogonal solution such that (6.1) holds if and only if the matrix 𝐻𝑁=^𝑇+𝐻2, (6.28) where 𝐻=𝐴^𝑇𝐵+𝐷𝐶^𝑇, (6.29) is invertible. Example 6.4 just illustrates it.
(2) The algorithm about computing the least squares skew-symmetric orthogonal solutions of the system (1.1) can be shown similarly; we omit it here.

7. Conclusions

This paper is devoted to giving the solvability conditions of and the expressions of the orthogonal solutions, the symmetric orthogonal solutions, and the skew-symmetric orthogonal solutions to the system (1.1), respectively, and meanwhile obtaining the least squares symmetric orthogonal and skew-symmetric orthogonal solutions of the system (1.1). In addition, an algorithm and an example have been provided to compute its least squares symmetric orthogonal solutions.

Acknowledgments

This research was supported by the grants from Innovation Program of Shanghai Municipal Education Commission (13ZZ080), the National Natural Science Foundation of China (11171205), the Natural Science Foundation of Shanghai (11ZR1412500), the Discipline Project at the corresponding level of Shanghai (A. 13-0101-12-005), and Shanghai Leading Academic Discipline Project (J50101).