On the Hermitian 𝑅-Conjugate Solution of a System of Matrix Equations

Abstract

Let 𝑅 be an 𝑛 by 𝑛 nontrivial real symmetric involution matrix, that is, 𝑅=𝑅⁻¹=𝑅^𝑇≠𝐼_𝑛. An 𝑛×𝑛 complex matrix 𝐴 is termed 𝑅-conjugate if 𝐴=𝑅𝐴𝑅, where 𝐴 denotes the conjugate of 𝐴. We give necessary and sufficient conditions for the existence of the Hermitian 𝑅-conjugate solution to the system of complex matrix equations 𝐴𝑋=𝐶and𝑋𝐵=𝐷 and present an expression of the Hermitian 𝑅-conjugate solution to this system when the solvability conditions are satisfied. In addition, the solution to an optimal approximation problem is obtained. Furthermore, the least squares Hermitian 𝑅-conjugate solution with the least norm to this system mentioned above is considered. The representation of such solution is also derived. Finally, an algorithm and numerical examples are given.

1. Introduction

Throughout, we denote the complex 𝑚×𝑛 matrix space by ℂ^𝑚×𝑛, the real 𝑚×𝑛 matrix space by ℝ^𝑚×𝑛, and the set of all matrices in ℝ^𝑚×𝑛 with rank 𝑟 by ℝ_𝑟^𝑚×𝑛. The symbols 𝐼,𝐴,𝐴^𝑇,𝐴^∗,𝐴^†, and ‖𝐴‖ stand for the identity matrix with the appropriate size, the conjugate, the transpose, the conjugate transpose, the Moore-Penrose generalized inverse, and the Frobenius norm of 𝐴∈ℂ^𝑚×𝑛, respectively. We use 𝑉_𝑛 to denote the 𝑛×𝑛 backward matrix having the elements 1 along the southwest diagonal and with the remaining elements being zeros.

Recall that an 𝑛×𝑛 complex matrix 𝐴 is centrohermitian if 𝐴=𝑉_𝑛𝐴𝑉_𝑛. Centrohermitian matrices and related matrices, such as k-Hermitian matrices, Hermitian Toeplitz matrices, and generalized centrohermitian matrices, appear in digital signal processing and others areas (see, [1–4]). As a generalization of a centrohermitian matrix and related matrices, Trench [5] gave the definition of 𝑅-conjugate matrix. A matrix 𝐴∈ℂ^𝑛×𝑛 is 𝑅-conjugate if 𝐴=𝑅𝐴𝑅, where 𝑅 is a nontrivial real symmetric involution matrix, that is, 𝑅=𝑅⁻¹=𝑅^𝑇 and 𝑅≠𝐼_𝑛. At the same time, Trench studied the linear equation 𝐴𝑧=𝑤 for 𝑅-conjugate matrices in [5], where 𝑧,𝑤 are known column vectors.

Investigating the matrix equation 𝐴𝑋=𝐵 (1.1) with the unknown matrix 𝑋 being symmetric, reflexive, Hermitian-generalized Hamiltonian, and repositive definite is a very active research topic [6–14]. As a generalization of (1.1), the classical system of matrix equations 𝐴𝑋=𝐶,𝑋𝐵=𝐷 (1.2) has attracted many author’s attention. For instance, [15] gave the necessary and sufficient conditions for the consistency of (1.2), [16, 17] derived an expression for the general solution by using singular value decomposition of a matrix and generalized inverses of matrices, respectively. Moreover, many results have been obtained about the system (1.2) with various constraints, such as bisymmetric, Hermitian, positive semidefinite, reflexive, and generalized reflexive solutions (see, [18–28]). To our knowledge, so far there has been little investigation of the Hermitian 𝑅-conjugate solution to (1.2).

Motivated by the work mentioned above, we investigate Hermitian 𝑅-conjugate solutions to (1.2). We also consider the optimal approximation problem 𝑋−𝐸=min_{𝑋∈𝑆𝑋}‖𝑋−𝐸‖, (1.3) where 𝐸 is a given matrix in ℂ^𝑛×𝑛 and 𝑆_𝑋 the set of all Hermitian 𝑅-conjugate solutions to (1.2). In many cases the system (1.2) has not Hermitian 𝑅-conjugate solution. Hence, we need to further study its least squares solution, which can be described as follows: Let 𝑅𝐻ℂ^𝑛×𝑛 denote the set of all Hermitian 𝑅-conjugate matrices in ℂ^𝑛×𝑛: 𝑆_𝐿=𝑋∣min_{𝑋∈𝑅𝐻ℂ𝑛×𝑛}‖𝐴𝑋−𝐶‖²+‖𝑋𝐵−𝐷‖². (1.4) Find 𝑋∈ℂ^𝑛×𝑛 such that 𝑋=min_{𝑋∈𝑆𝐿}‖𝑋‖. (1.5)

In Section 2, we present necessary and sufficient conditions for the existence of the Hermitian 𝑅-conjugate solution to (1.2) and give an expression of this solution when the solvability conditions are met. In Section 3, we derive an optimal approximation solution to (1.3). In Section 4, we provide the least squares Hermitian 𝑅-conjugate solution to (1.5). In Section 5, we give an algorithm and a numerical example to illustrate our results.

2. 𝑅-Conjugate Hermitian Solution to (1.2)

In this section, we establish the solvability conditions and the general expression for the Hermitian 𝑅-conjugate solution to (1.2).

We denote 𝑅ℂ^𝑛×𝑛 and 𝑅𝐻ℂ^𝑛×𝑛 the set of all 𝑅-conjugate matrices and Hermitian 𝑅-conjugate matrices, respectively, that is, 𝑅ℂ^𝑛×𝑛=𝐴∣𝐴=𝑅𝐴𝑅,𝐻𝑅ℂ^𝑛×𝑛=𝐴∣𝐴=𝑅𝐴𝑅,𝐴=𝐴^∗, (2.1) where 𝑅 is 𝑛×𝑛 nontrivial real symmetric involution matrix.

Chang et al. in [29] mentioned that for nontrivial symmetric involution matrix 𝑅∈ℝ^𝑛×𝑛, there exist positive integer 𝑟 and 𝑛×𝑛 real orthogonal matrix [𝑃,𝑄] such that 𝐼𝑅=𝑃,𝑄^𝑟00−𝐼_𝑛−𝑟𝑃^𝑇𝑄^𝑇, (2.2) where 𝑃∈ℝ^𝑛×𝑟,𝑄∈ℝ^{𝑛×(𝑛−𝑟)}. By (2.2), 𝑅𝑃=𝑃,𝑅𝑄=−𝑄,𝑃^𝑇𝑃=𝐼_𝑟,𝑄^𝑇𝑄=𝐼_𝑛−𝑟,𝑃^𝑇𝑄=0,𝑄^𝑇𝑃=0. (2.3)

Throughout this paper, we always assume that the nontrivial symmetric involution matrix 𝑅 is fixed which is given by (2.2) and (2.3). Now, we give a criterion of judging a matrix is 𝑅-conjugate Hermitian matrix.

Theorem 2.1. A matrix 𝐾∈𝐻𝑅ℂ^𝑛×𝑛 if and only if there exists a symmetric matrix 𝐻∈ℝ^𝑛×𝑛 such that 𝐾=Γ𝐻Γ^∗, where Γ=𝑃,𝑖𝑄, (2.4) with 𝑃,𝑄 being the same as (2.2).

Proof. If 𝐾∈𝐻𝑅ℂ^𝑛×𝑛, then 𝐾=𝑅𝐾𝑅. By (2.2), 𝐼𝐾=𝑅𝐾𝑅=𝑃,𝑄^𝑟00−𝐼_𝑛−𝑟𝑃^𝑇𝑄^𝑇𝐾𝐼𝑃,𝑄^𝑟00−𝐼_𝑛−𝑟𝑃^𝑇𝑄^𝑇, (2.5) which is equivalent to 𝑃^𝑇𝑄^𝑇𝐾=𝐼𝑃,𝑄_𝑟00−𝐼_𝑛−𝑟𝑃^𝑇𝑄^𝑇𝐾𝐼𝑃,𝑄^𝑟00−𝐼_𝑛−𝑟. (2.6) Suppose that 𝑃^𝑇𝑄^𝑇𝐾=𝐾𝑃,𝑄¹¹𝐾₁₂𝐾₂₁𝐾₂₂. (2.7) Substituting (2.7) into (2.6), we obtain 𝐾₁₁𝐾₁₂𝐾₂₁𝐾₂₂=𝐼_𝑟00−𝐼_𝑛−𝑟𝐾₁₁𝐾₁₂𝐾₂₁𝐾₂₂𝐼_𝑟00−𝐼_𝑛−𝑟=𝐾₁₁−𝐾₁₂−𝐾₂₁𝐾₂₂. (2.8) Hence, 𝐾₁₁=𝐾₁₁,𝐾₁₂=−𝐾₁₂,𝐾₂₁=−𝐾₂₁,𝐾₂₂=𝐾₂₂,thatis,𝐾₁₁,𝑖𝐾₁₂,𝑖𝐾₂₁,𝐾₂₂ are real matrices. If we denote 𝑀=𝑖𝐾₁₂,𝑁=−𝑖𝐾₂₁, then by (2.7) 𝐾𝐾=𝑃,𝑄¹¹𝐾₁₂𝐾₂₁𝐾₂₂𝑃^𝑇𝑄^𝑇=𝐾𝑃,𝑖𝑄¹¹𝑀𝑁𝐾₂₂𝑃^𝑇−𝑖𝑄^𝑇. (2.9) Let Γ=[𝑃,𝑖𝑄], and 𝐾𝐻=¹¹𝑀𝑁𝐾₂₂. (2.10) Then, 𝐾 can be expressed as Γ𝐻Γ^∗, where Γ is unitary matrix and 𝐻 is a real matrix. By 𝐾=𝐾^∗Γ𝐻^𝑇Γ^∗=𝐾^∗=𝐾=Γ𝐻Γ^∗, (2.11) we obtain 𝐻=𝐻^𝑇.
Conversely, if there exists a symmetric matrix 𝐻∈ℝ^𝑛×𝑛 such that 𝐾=Γ𝐻Γ^∗, then it follows from (2.3) that 𝑅𝐾𝑅=𝑅Γ𝐻Γ^∗𝐻𝑃𝑅=𝑅𝑃,𝑖𝑄^𝑇−𝑖𝑄^𝑇𝐻𝑃𝑅=𝑃,−𝑖𝑄^𝑇𝑖𝑄^𝑇=Γ𝐻Γ^∗=𝐾𝐾,_∗=Γ𝐻^𝑇Γ^∗=Γ𝐻Γ^∗=𝐾, (2.12) that is, 𝐾∈𝐻𝑅ℂ^𝑛×𝑛.

Theorem 2.1 implies that an arbitrary complex Hermitian 𝑅-conjugate matrix is equivalent to a real symmetric matrix.

Lemma 2.2. For any matrix 𝐴∈ℂ^𝑚×𝑛,𝐴=𝐴₁+𝑖𝐴₂, where 𝐴₁=𝐴+𝐴2,𝐴₂=𝐴−𝐴.2𝑖 (2.13)

Proof. For any matrix 𝐴∈ℂ^𝑚×𝑛, it is obvious that 𝐴=𝐴₁+𝑖𝐴₂, where 𝐴₁,𝐴₂ are defined as (2.13). Now, we prove that the decomposition 𝐴=𝐴₁+𝑖𝐴₂ is unique. If there exist 𝐵₁,𝐵₂ such that 𝐴=𝐵₁+𝑖𝐵₂, then 𝐴₁−𝐵₁+𝑖𝐵₂−𝐴₂=0. (2.14) It follows from 𝐴₁,𝐴₂,𝐵₁,and𝐵₂ are real matrix that 𝐴₁=𝐵₁,𝐴₂=𝐵₂. (2.15) Hence, 𝐴=𝐴₁+𝑖𝐴₂ holds, where 𝐴₁,𝐴₂ are defined as (2.13).

By Theorem 2.1, for 𝑋∈𝐻𝑅ℂ^𝑛×𝑛, we may assume that 𝑋=Γ𝑌Γ^∗, (2.16) where Γ is defined as (2.4) and 𝑌∈ℝ^𝑛×𝑛 is a symmetric matrix.

Suppose that 𝐴Γ=𝐴₁+𝑖𝐴₂∈ℂ^𝑚×𝑛,𝐶Γ=𝐶₁+𝑖𝐶₂∈ℂ^𝑚×𝑛,Γ^∗𝐵=𝐵₁+𝑖𝐵₂∈ℂ^𝑛×𝑙, and Γ^∗𝐷=𝐷₁+𝑖𝐷₂∈ℂ^𝑛×𝑙, where 𝐴₁=𝐴Γ+𝐴Γ2,𝐴₂=𝐴Γ−𝐴Γ2𝑖,𝐶₁=𝐶Γ+𝐶Γ2,𝐶₂=𝐶Γ−𝐶Γ,𝐵2𝑖₁=Γ^∗𝐵+Γ^∗𝐵2,𝐵₂=Γ^∗𝐵−Γ^∗𝐵2𝑖,𝐷₁=Γ^∗𝐷+Γ^∗𝐷2,𝐷₂=Γ^∗𝐷−Γ^∗𝐷.2𝑖 (2.17) Then, system (1.2) can be reduced into 𝐴₁+𝑖𝐴₂𝑌=𝐶₁+𝑖𝐶₂,𝑌𝐵₁+𝑖𝐵₂=𝐷₁+𝑖𝐷₂, (2.18) which implies that 𝐴₁𝐴₂𝐶𝑌=¹𝐶₂,𝑌𝐵₁,𝐵₂=𝐷₁,𝐷₂. (2.19) Let 𝐴𝐹=¹𝐴₂𝐶,𝐺=¹𝐶₂,𝐾=𝐵₁,𝐵₂,𝐿=𝐷₁,𝐷₂𝐹𝐾,𝑀=_𝑇𝐺𝐿,𝑁=_𝑇. (2.20) Then, system (1.2) has a solution 𝑋 in 𝐻𝑅ℂ^𝑛×𝑛 if and only if the real system 𝑀𝑌=𝑁 (2.21) has a symmetric solution 𝑌 in ℝ^𝑛×𝑛.

Lemma 2.3 (Theorem  1 in [7]). Let 𝐴∈ℝ^𝑚×𝑛. The 𝑆𝑉𝐷 of matrix 𝐴 is as follows 𝐴=𝑈𝑉Σ000^𝑇, (2.22) where 𝑈=[𝑈₁,𝑈₂]∈ℝ^𝑚×𝑚 and 𝑉=[𝑉₁,𝑉₂]∈ℝ^𝑛×𝑛 are orthogonal matrices, Σ=diag(𝜎₁,…,𝜎_𝑟),𝜎_𝑖>0(𝑖=1,…,𝑟),𝑟=rank(𝐴),𝑈₁∈ℝ^𝑚×𝑟,𝑉₁∈ℝ^𝑛×𝑟. Then, (1.1) has a symmetric solution if and only if 𝐴𝐵^𝑇=𝐵𝐴^𝑇,𝑈^𝑇₂𝐵=0. (2.23) In that case, it has the general solution 𝑋=𝑉₁Σ⁻¹𝑈^𝑇₁𝐵+𝑉₂𝑉^𝑇₂𝐵^𝑇𝑈₁Σ⁻¹𝑉^𝑇₁+𝑉₂𝐺𝑉^𝑇₂, (2.24) where 𝐺 is an arbitrary (𝑛−𝑟)×(𝑛−𝑟) symmetric matrix.

By Lemma 2.3, we have the following theorem.

Theorem 2.4. Given 𝐴∈ℂ^𝑚×𝑛,𝐶∈ℂ^𝑚×𝑛,𝐵∈ℂ^𝑛×𝑙, and 𝐷∈ℂ^𝑛×𝑙. Let 𝐴₁,𝐴₂,𝐶₁,𝐶₂,𝐵₁,𝐵₂,𝐷₁,𝐷₂,𝐹,𝐺,𝐾,𝐿,𝑀,𝑎𝑛𝑑𝑁 be defined in (2.17), (2.20), respectively. Assume that the 𝑆𝑉𝐷 of 𝑀∈ℝ^{(2𝑚+2𝑙)×𝑛} is as follows 𝑀𝑀=𝑈¹0𝑉00^𝑇, (2.25) where 𝑈=[𝑈₁,𝑈₂]∈ℝ^{(2𝑚+2𝑙)×(2𝑚+2𝑙)} and 𝑉=[𝑉₁,𝑉₂]∈ℝ^𝑛×𝑛 are orthogonal matrices, 𝑀₁=diag(𝜎₁,…,𝜎_𝑟),𝜎_𝑖>0(𝑖=1,…,𝑟),𝑟=rank(𝑀),𝑈₁∈ℝ^{(2𝑚+2𝑙)×𝑟},𝑉₁∈ℝ^𝑛×𝑟. Then, system (1.2) has a solution in 𝐻𝑅ℂ^𝑛×𝑛 if and only if 𝑀𝑁^𝑇=𝑁𝑀^𝑇,𝑈^𝑇₂𝑁=0. (2.26) In that case, it has the general solution 𝑋=Γ𝑉₁𝑀₁⁻¹𝑈^𝑇₁𝑁+𝑉₂𝑉^𝑇₂𝑁^𝑇𝑈₁𝑀₁⁻¹𝑉^𝑇₁+𝑉₂𝐺𝑉^𝑇₂Γ^∗, (2.27) where 𝐺 is an arbitrary (𝑛−𝑟)×(𝑛−𝑟) symmetric matrix.

3. The Solution of Optimal Approximation Problem (1.3)

When the set 𝑆_𝑋 of all Hermitian 𝑅-conjugate solution to (1.2) is nonempty, it is easy to verify 𝑆_𝑋 is a closed set. Therefore, the optimal approximation problem (1.3) has a unique solution by [30].

Theorem 3.1. Given 𝐴∈ℂ^𝑚×𝑛,𝐶∈ℂ^𝑚×𝑛,𝐵∈ℂ^𝑛×𝑙,𝐷∈ℂ^𝑛×𝑙,𝐸∈ℂ^𝑛×𝑛, and 𝐸₁=(1/2)(Γ^∗𝐸Γ+Γ^∗𝐸Γ). Assume 𝑆_𝑋 is nonempty, then the optimal approximation problem (1.3) has a unique solution 𝑋 and 𝑋=Γ𝑉₁𝑀₁⁻¹𝑈^𝑇₁𝑁+𝑉₂𝑉^𝑇₂𝑁^𝑇𝑈₁𝑀₁⁻¹𝑉^𝑇₁+𝑉₂𝑉^𝑇₂𝐸₁𝑉₂𝑉^𝑇₂Γ^∗. (3.1)

Proof. Since 𝑆_𝑋 is nonempty, 𝑋∈𝑆_𝑋 has the form of (2.27). By Lemma 2.2, Γ^∗𝐸Γ can be written as Γ^∗𝐸Γ=𝐸₁+𝑖𝐸₂, (3.2) where 𝐸₁=12Γ^∗𝐸Γ+Γ^∗𝐸Γ,𝐸₂=1Γ2𝑖^∗𝐸Γ−Γ^∗𝐸Γ. (3.3) According to (3.2) and the unitary invariance of Frobenius norm ‖𝑋−𝐸‖=Γ𝑉₁𝑀⁻¹𝑈^𝑇₁𝑁+𝑉₂𝑉^𝑇₂𝑁^𝑇𝑈₁𝑀⁻¹𝑉^𝑇₁+𝑉₂𝐺𝑉^𝑇₂Γ^∗=𝐸−𝐸₁−𝑉₁𝑀⁻¹𝑈^𝑇₁𝑁−𝑉₂𝑉^𝑇₂𝑁^𝑇𝑈₁𝑀⁻¹𝑉^𝑇₁−𝑉₂𝐺𝑉^𝑇₂+𝑖𝐸₂. (3.4) We get ‖𝑋−𝐸‖²=𝐸₁−𝑉₁𝑀⁻¹𝑈^𝑇₁𝑁−𝑉₂𝑉^𝑇₂𝑁^𝑇𝑈₁𝑀⁻¹𝑉^𝑇₁−𝑉₂𝐺𝑉^𝑇₂²+𝐸₂². (3.5) Then, min_{𝑋∈𝑆𝑋}‖𝑋−𝐸‖ is consistent if and only if there exists 𝐺∈ℝ^{(𝑛−𝑟)×(𝑛−𝑟)} such that min𝐸₁−𝑉₁𝑀⁻¹𝑈^𝑇₁𝑁−𝑉₂𝑉^𝑇₂𝑁^𝑇𝑈₁𝑀⁻¹𝑉^𝑇₁−𝑉₂𝐺𝑉^𝑇₂. (3.6) For the orthogonal matrix 𝑉𝐸₁−𝑉₁𝑀⁻¹𝑈^𝑇₁𝑁−𝑉₂𝑉^𝑇₂𝑁^𝑇𝑈₁𝑀⁻¹𝑉^𝑇₁−𝑉₂𝐺𝑉^𝑇₂²=𝑉^𝑇(𝐸₁−𝑉₁𝑀⁻¹𝑈^𝑇₁𝑁−𝑉₂𝑉^𝑇₂𝑁^𝑇𝑈₁𝑀⁻¹𝑉^𝑇₁−𝑉₂𝐺𝑉^𝑇₂)𝑉²=𝑉^𝑇₁𝐸₁−𝑉₁𝑀⁻¹𝑈^𝑇₁𝑁𝑉₁²+𝑉^𝑇₁(𝐸₁−𝑉₁𝑀⁻¹𝑈^𝑇₁𝑁)𝑉₂²+𝑉^𝑇₂𝐸₁−𝑉₂𝑉^𝑇₂𝑁^𝑇𝑈₁𝑀⁻¹𝑉^𝑇₁𝑉₁²+𝑉^𝑇₂𝐸₁−𝑉₂𝐺𝑉^𝑇₂𝑉₂. (3.7) Therefore, min𝐸₁−𝑉₁𝑀⁻¹𝑈^𝑇₁𝑁−𝑉₂𝑉^𝑇₂𝑁^𝑇𝑈₁𝑀⁻¹𝑉^𝑇₁−𝑉₂𝐺𝑉^𝑇₂ (3.8) is equivalent to 𝐺=𝑉^𝑇₂𝐸₁𝑉₂. (3.9) Substituting (3.9) into (2.27), we obtain (3.1).

4. The Solution of Problem (1.5)

In this section, we give the explicit expression of the solution to (1.5).

Theorem 4.1. Given 𝐴∈ℂ^𝑚×𝑛,𝐶∈ℂ^𝑚×𝑛,𝐵∈ℂ^𝑛×𝑙,  and  𝐷∈ℂ^𝑛×𝑙. Let 𝐴₁,𝐴₂,𝐶₁,𝐶₂,𝐵₁,𝐵₂,𝐷₁,𝐷₂,𝐹,𝐺,𝐾,𝐿,𝑀,  and  𝑁 be defined in (2.17), (2.20), respectively. Assume that the 𝑆𝑉𝐷 of 𝑀∈ℝ^{(2𝑚+2𝑙)×𝑛} is as (2.25) and system (1.2) has not a solution in 𝐻𝑅ℂ^𝑛×𝑛. Then, 𝑋∈𝑆_𝐿 can be expressed as 𝑀𝑋=Γ𝑉¹⁻¹𝑈^𝑇₁𝑁𝑉₁𝑀₁⁻¹𝑈^𝑇₁𝑁𝑉₂𝑉^𝑇₂𝑁^𝑇𝑈₁𝑀₁⁻¹𝑌₂₂𝑉^𝑇Γ^∗, (4.1) where 𝑌₂₂∈ℝ^{(𝑛−𝑟)×(𝑛−𝑟)} is an arbitrary symmetric matrix.

Proof. It yields from (2.17)–(2.21) and (2.25) that ‖𝐴𝑋−𝐶‖²+‖𝑋𝐵−𝐷‖²=‖𝑀𝑌−𝑁‖²=𝑈𝑀₁0𝑉00^𝑇𝑌−𝑁²=𝑀₁0𝑉00^𝑇𝑌𝑉−𝑈^𝑇𝑁𝑉². (4.2) Assume that 𝑉^𝑇𝑌𝑌𝑉=¹¹𝑌₁₂𝑌₂₁𝑌₂₂,𝑌₁₁∈ℝ^𝑟×𝑟,𝑌₂₂∈ℝ^{(𝑛−𝑟)×(𝑛−𝑟)}. (4.3) Then, we have ‖𝐴𝑋−𝐶‖²+‖𝑋𝐵−𝐷‖²=𝑀₁𝑌₁₁−𝑈^𝑇₁𝑁𝑉₁²+𝑀₁𝑌₁₂−𝑈^𝑇₁𝑁𝑉₂²+𝑈^𝑇₂𝑁𝑉₁²+𝑈^𝑇₂𝑁𝑉₂². (4.4) Hence, min‖𝐴𝑋−𝐶‖²+‖𝑋𝐵−𝐷‖² (4.5) is solvable if and only if there exist 𝑌₁₁,𝑌₁₂ such that 𝑀₁𝑌₁₁−𝑈^𝑇₁𝑁𝑉₁²𝑀=min,₁𝑌₁₂−𝑈^𝑇₁𝑁𝑉₂²=min. (4.6) It follows from (4.6) that 𝑌₁₁=𝑀₁⁻¹𝑈^𝑇₁𝑁𝑉₁,𝑌₁₂=𝑀₁⁻¹𝑈^𝑇₁𝑁𝑉₂. (4.7) Substituting (4.7) into (4.3) and then into (2.16), we can get that the form of elements in 𝑆_𝐿 is (4.1).

Theorem 4.2. Assume that the notations and conditions are the same as Theorem 4.1. Then, 𝑋=min_{𝑋∈𝑆𝐿}‖𝑋‖ (4.8) if and only if 𝑀𝑋=Γ𝑉¹⁻¹𝑈^𝑇₁𝑁𝑉₁𝑀₁⁻¹𝑈^𝑇₁𝑁𝑉₂𝑉^𝑇₂𝑁^𝑇𝑈₁𝑀₁⁻¹0𝑉^𝑇Γ^∗. (4.9)

Proof. In Theorem 4.1, it implies from (4.1) that min_{𝑋∈𝑆𝐿}‖𝑋‖ is equivalent to 𝑋 has the expression (4.1) with 𝑌₂₂=0. Hence, (4.9) holds.

5. An Algorithm and Numerical Example

Base on the main results of this paper, we in this section propose an algorithm for finding the solution of the approximation problem (1.3) and the least squares problem with least norm (1.5). All the tests are performed by MATLAB 6.5 which has a machine precision of around 10⁻¹⁶.

Algorithm 5.1. (1) Input 𝐴∈ℂ^𝑚×𝑛,𝐶∈ℂ^𝑚×𝑛,𝐵∈ℂ^𝑛×𝑙,𝐷∈ℂ^𝑛×𝑙.
(2) Compute 𝐴₁,𝐴₂,𝐶₁,𝐶₂,𝐵₁,𝐵₂,𝐷₁,𝐷₂,𝐹,𝐺,𝐾,𝐿,𝑀,and𝑁 by (2.17) and (2.20).
(3) Compute the singular value decomposition of 𝑀 with the form of (2.25).
(4) If (2.26) holds, then input 𝐸∈ℂ^𝑛×𝑛 and compute the solution 𝑋 of problem (1.3) according (3.1), else compute the solution 𝑋 to problem (1.5) by (4.9).

To show our algorithm is feasible, we give two numerical example. Let an nontrivial symmetric involution be 𝑅=.10000−1000010000−1 (5.1) We obtain [𝑃,𝑄] in (2.2) by using the spectral decomposition of 𝑅, then by (2.4) Γ=.100000𝑖00100000𝑖 (5.2)

Example 5.2. Suppose 𝐴∈ℂ^2×4,𝐶∈ℂ^2×4,𝐵∈ℂ^4×3,𝐷∈ℂ^4×3, and ,,,.𝐴=3.33−5.987𝑖45𝑖7.21−𝑖0−0.66𝑖7.6941.123𝑖𝐶=0.2679−0.0934𝑖0.0012+4.0762𝑖−0.0777−0.1718𝑖−1.2801𝑖0.2207−0.1197𝑖0.08770.7058𝑖𝐵=4+12𝑖2.369𝑖4.256−5.111𝑖4𝑖4.66𝑖8.21−5𝑖04.83𝑖56+𝑖2.22𝑖−4.6667𝑖𝐷=0.0616+0.1872𝑖−0.0009+0.1756𝑖1.6746−0.0494𝑖0.0024+0.2704𝑖0.1775+0.4194𝑖0.7359−0.6189𝑖−0.0548+0.3444𝑖0.0093−0.3075𝑖−0.4731−0.1636𝑖0.0337𝑖0.1209−0.1864𝑖−0.2484−3.8817𝑖 (5.3) We can verify that (2.26) holds. Hence, system (1.2) has an Hermitian 𝑅-conjugate solution. Given 𝐸=.7.35𝑖8.389𝑖99.256−6.51𝑖−4.6𝑖1.554.56𝑖7.71−7.5𝑖𝑖5𝑖0−4.556𝑖−7.994.22𝑖05.1𝑖0 (5.4) Applying Algorithm 5.1, we obtain the following: 𝑋=.1.55970.0194𝑖2.87050.0002𝑖−0.0194𝑖9.00010.2005𝑖−3.99972.8705−0.2005𝑖−0.04527.9993𝑖−0.0002𝑖−3.9997−7.9993𝑖5.6846 (5.5)

Example 5.2 illustrates that we can solve the optimal approximation problem with Algorithm 5.1 when system (1.2) have Hermitian 𝑅-conjugate solutions.

Example 5.3. Let 𝐴,𝐵,and𝐶 be the same as Example 5.2, and let 𝐷 in Example 5.2 be changed into 𝐷=.0.0616+0.1872𝑖−0.0009+0.1756𝑖1.6746+0.0494𝑖0.0024+0.2704𝑖0.1775+0.4194𝑖0.7359−0.6189𝑖−0.0548+0.3444𝑖0.0093−0.3075𝑖−0.4731−0.1636𝑖0.0337𝑖0.1209−0.1864𝑖−0.2484−3.8817𝑖 (5.6) We can verify that (2.26) does not hold. By Algorithm 5.1, we get 𝑋=.0.522.2417𝑖0.49140.3991𝑖−2.2417𝑖8.66340.1921𝑖−2.82320.4914−0.1921𝑖0.14061.3154𝑖−0.3991𝑖−2.8232−1.3154𝑖6.3974 (5.7)

Example 5.3 demonstrates that we can get the least squares solution with Algorithm 5.1 when system (1.2) has not Hermitian 𝑅-conjugate solutions.

Acknowledgments

This research was supported by the Grants from the Key Project of Scientific Research Innovation Foundation of Shanghai Municipal Education Commission (13ZZ080), the National Natural Science Foundation of China (11171205), the Natural Science Foundation of Shanghai (11ZR1412500), the Ph.D. Programs Foundation of Ministry of Education of China (20093108110001), the Discipline Project at the corresponding level of Shanghai (A. 13-0101-12-005), Shanghai Leading Academic Discipline Project (J50101), the Natural Science Foundation of Hebei province (A2012403013), and the Natural Science Foundation of Hebei province (A2012205028). The authors are grateful to the anonymous referees for their helpful comments and constructive suggestions.