(Anti-)Hermitian Generalized (Anti-)Hamiltonian Solution to a System of Matrix Equations

Abstract

We mainly solve three problems. Firstly, by the decomposition of the (anti-)Hermitian generalized (anti-)Hamiltonian matrices, the necessary and sufficient conditions for the existence of and the expression for the (anti-)Hermitian generalized (anti-)Hamiltonian solutions to the system of matrix equations 𝐴𝑋=𝐵,𝑋𝐶=𝐷 are derived, respectively. Secondly, the optimal approximation solution min_𝑋∈𝐾‖𝑋−𝑋‖ is obtained, where 𝐾 is the (anti-)Hermitian generalized (anti-)Hamiltonian solution set of the above system and 𝑋 is the given matrix. Thirdly, the least squares (anti-)Hermitian generalized (anti-)Hamiltonian solutions are considered. In addition, algorithms about computing the least squares (anti-)Hermitian generalized (anti-)Hamiltonian solution and the corresponding numerical examples are presented.

1. Introduction

Throughout this paper, the set of all 𝑚×𝑛 complex matrices, the set of all 𝑛×𝑛 Hermitian matrices, the set of all 𝑛×𝑛 anti-Hermitian matrices, the set of all 𝑛×𝑛 unitary matrices, and the set of all 𝑛×𝑛 antisymmetric orthogonal matrices are denoted, respectively, by ℂ^𝑚×𝑛, 𝐻ℂ^𝑛×𝑛,𝐴𝐻ℂ^𝑛×𝑛,𝑈ℂ^𝑛×𝑛, and 𝐴𝑆𝑂ℝ^𝑛×𝑛. The symbol 𝐼_𝑛 represents an identity matrix of order 𝑛 and 𝑟(𝐴),𝐴^†, and 𝐴^∗, respectively, stand for the rank, the Moore-Penrose inverse, and the conjugate transpose of matrix 𝐴. For two matrices 𝐴,𝐵∈ℂ^𝑚×𝑛, the inner product is defined by ⟨𝐴,𝐵⟩=tr(𝐵^∗𝐴). Obviously, ℂ^𝑚×𝑛 is a complete inner product space. The norm ‖⋅‖, induced by the inner product, is called the Frobenius norm. 𝐴∗𝐵 stands for the Hadamard product of two matrices 𝐴 and 𝐵. For 𝐴∈ℂ^𝑚×𝑛, two matrices 𝐿_𝐴  and   𝑅_𝐴, respectively, represent two orthogonal projectors 𝐿_𝐴=𝐼_𝑛−𝐴^†𝐴 and 𝑅_𝐴=𝐼_𝑚−𝐴𝐴^†, both of which satisfy 𝐿_𝐴=𝐿_𝐴²=𝐿_𝐴^∗=𝐿_𝐴^†,𝑅_𝐴=𝑅_𝐴²=𝑅_𝐴^∗=𝑅_𝐴^†. (1)

The Hamiltonian matrices defined as in [1] are very important in engineering (see [2] and the references therein). Moreover, using Hamiltonian matrices to solve algebraic matrix Riccati equation is a very effective method in optimal control theory [3–5]. As the extension of the Hamiltonian matrices, the following four definitions, which can also be found in [1, 6, 7], are given. Without special statement, we in this paper always assume that 𝐽∈𝐴𝑆𝑂ℝ^2𝑘×2𝑘 satisfies 𝐽^𝑇=−𝐽,𝐽^𝑇𝐽=𝐽𝐽^𝑇=𝐼_𝑛. (2)

Definition 1. A matrix 𝑋∈𝐻𝐻ℂ^2𝑘×2𝑘 is said to be a Hermitian generalized Hamiltonian matrix if 𝑋=𝑋^∗ and 𝐽𝑋𝐽=𝑋^∗.

Definition 2. A matrix 𝑋∈𝐻𝐴𝐻ℂ^2𝑘×2𝑘 is said to be a Hermitian generalized anti-Hamiltonian matrix if 𝑋=𝑋^∗ and 𝐽𝑋𝐽=−𝑋^∗.

Definition 3. A matrix 𝑋∈𝐴𝐻𝐴𝐻ℂ^2𝑘×2𝑘 is said to be an anti-Hermitian generalized anti-Hamiltonian matrix if 𝑋=−𝑋^∗ and 𝐽𝑋𝐽=−𝑋^∗.

Definition 4. A matrix 𝑋∈𝐴𝐻𝐻ℂ^2𝑘×2𝑘 is said to be an anti-Hermitian generalized Hamiltonian matrix if 𝑋=−𝑋^∗ and 𝐽𝑋𝐽=𝑋^∗.

The well-known system of matrix equations 𝐴𝑋=𝐵,𝑋𝐶=𝐷, (3) with unknown matrix 𝑋, has attracted much attention and has been widely and deeply studied by many authors. For example, Khatri and Mitra [8] in 1976 established the Hermitian and nonnegative definite solution to the system (3). Mitra [9] in 1984 gave the system (3) the minimal rank solution over the complex field ℂ. Wang in [10] and Wang et al. [11], respectively, investigated the bisymmetric and centrosymmetric solutions over the quaternion algebra and obtained the bisymmetric nonnegative definite solutions with extremal ranks and inertias to the system (3). Xu in [12] studied the common Hermitian and positive solutions to the adjointable operator equations (3). Yuan in [13] presented the least squares solutions to the system (3). Some other results concerning the system (3) can be found in [14–23].

As special cases of the system (3), the classical matrix equations 𝐴𝑋=𝐵 and 𝑋𝐶=𝐷 have also been investigated (see, e.g., [1, 2, 5–7, 24–31]). For instance, Dai [24], by means of the singular value decomposition, derived the symmetric solution to equation 𝐴𝑋=𝐵. Guan and Jiang [6], using the decomposition of the anti-Hermitian generalized anti-Hamiltonian matrices, derived the least squares solution to equation 𝐴𝑋=𝐵. Zhang et al. in [29] and [1], respectively, obtained the general expression of the least squares Hermitian generalized Hamiltonian solutions to equation 𝑋𝐶=𝐷 and got the unite optimal approximation solution in the least squares solutions set and gave the solvable conditions and the general representation of the Hermitian generalized Hamiltonian solutions to equation 𝐴𝑋=𝐵, by using the singular value decomposition and the properties of Hermitian generalized Hamiltonian matrices.

As far as we know, there has been little information on studying the (anti-)Hermitian generalized (anti-)Hamiltonian solution to the system (3) over ℂ^2𝑘×2𝑘. So, motived by the work mentioned above, especially the work in [6, 7, 26, 29, 30], we, in this paper, are mainly concerned with the following three problems.

Problem 5. Given 𝐴,𝐵∈ℂ^𝑚×2𝑘,𝐶,𝐷∈ℂ^2𝑘×𝑞, find 𝑋∈𝐻𝐻ℂ^2𝑘×2𝑘(𝐻𝐴𝐻ℂ^2𝑘×2𝑘,𝐴𝐻𝐻ℂ^2𝑘×2𝑘, or 𝐴𝐻𝐴𝐻ℂ^2𝑘×2𝑘) such that the system (3) holds.

Problem 6. Given 𝑋∈ℂ^2𝑘×2𝑘, find 𝑋∈𝐾 such that 𝑋−𝑋=min_𝑋∈𝐾𝑋−𝑋, (4) where 𝐾 is the solution set of Problem 5.

Problem 7. Let 𝐴,𝐵∈ℂ^𝑚×2𝑘,𝐶,𝐷∈ℂ^2𝑘×𝑞. Find 𝑋∈𝐻𝐻ℂ^2𝑘×2𝑘(𝐻𝐴𝐻ℂ^2𝑘×2𝑘,𝐴𝐻𝐻ℂ^2𝑘×2𝑘, or 𝐴𝐻𝐴𝐻ℂ^2𝑘×2𝑘) such that min_𝑋=‖𝐴𝑋−𝐵‖²+‖𝑋𝐶−𝐷‖². (5)

The remainder of this paper is arranged as follows. In Section 2, some lemmas will be introduced, which will be useful for us to obtain the solutions to Problems 5–7. In Section 3, by applying the decomposition of the (anti-)Hermitian generalized (anti-)Hamiltonian matrices, the solvability condition and the explicit expression of the solution to Problem 5 will be derived. In Section 4, the optimal approximation solution to Problem 6 will be established. In Section 5, the solution to Problem 7 will be investigated and meanwhile the minimum norm of the solution will be obtained. In Section 6, algorithms and numerical examples about computing the solution to Problem 7 will be provided. Finally, in Section 7, some conclusions will be made.

2. Preliminaries

In this section, we focus on introducing some lemmas, which will play key roles in solving Problems 5–7.

Taking into account Definitions 1–4 and the eigenvalue decomposition of the matrix 𝐽∈𝐴𝑆𝑂ℝ^2𝑘×2𝑘, it is not difficult to conclude that the following decompositions of the (anti-)Hermitian generalized (anti-)Hamiltonian matrices hold, some of which can also be seen in [6, 26, 29, 30].

Lemma 8. Let the eigenvalue decomposition of matrix 𝐽∈𝐴𝑆𝑂ℝ^2𝑘×2𝑘 be 𝐽=𝑃𝑖𝐼_𝑘00−𝑖𝐼_𝑘𝑃^∗, (6) where 𝑃∈𝑈ℂ^2𝑘×2𝑘. Then 𝑋∈𝐻𝐻ℂ^2𝑘×2𝑘 if and only if 𝑋 can be expressed as 𝑋=𝑃0𝑋₁₂𝑋^∗₁₂0𝑃^∗, (7) where 𝑋₁₂∈ℂ^𝑘×𝑘 are arbitrary.

Lemma 9. Let the eigenvalue decomposition of matrix 𝐽∈𝐴𝑆𝑂ℝ^2𝑘×2𝑘 be (6). Then 𝑋∈𝐴𝐻𝐴𝐻ℂ^2𝑘×2𝑘 if and only if 𝑋 can be expressed as 𝑋=𝑃0𝑋₁₂−𝑋^∗₁₂0𝑃^∗, (8) where 𝑋₁₂∈ℂ^𝑘×𝑘 is arbitrary.

Lemma 10. Let the eigenvalue decomposition of matrix 𝐽∈𝐴𝑆𝑂ℝ^2𝑘×2𝑘 be (6). Then 𝑋∈𝐻𝐴𝐻ℂ^2𝑘×2𝑘 if and only if 𝑋 can be expressed as 𝑋𝑋=𝑃¹¹00𝑋₂₂𝑃^∗, (9) where 𝑋₁₁,𝑋₂₂∈𝐻ℂ^𝑘×𝑘 are arbitrary.

Lemma 11. Let the eigenvalue decomposition of matrix 𝐽∈𝐴𝑆𝑂ℝ^2𝑘×2𝑘 be (6). Then 𝑋∈𝐴𝐻𝐻ℂ^2𝑘×2𝑘 if and only if 𝑋 can be expressed as 𝑋𝑋=𝑃¹¹00𝑋₂₂𝑃^∗, (10) where 𝑋₁₁,𝑋₂₂∈𝐴𝐻ℂ^𝑘×𝑘 are arbitrary.

Lemma 12 (see [20]). Given 𝐴∈ℂ^𝑚×𝑛,𝐵∈ℂ^𝑝×𝑙,𝐶∈ℂ^𝑚×𝑝, and 𝐷∈ℂ^𝑛×𝑙, then the system of matrix equations 𝐴𝑋=𝐶,𝑋𝐵=𝐷 (11) has a solution 𝑋∈ℂ^𝑛×𝑝 if and only if 𝐴𝐴^†𝐶=𝐶,𝐷𝐵^†𝐵=𝐷,𝐴𝐷=𝐶𝐵, (12) in which case the general solutions can be expressed as 𝑋=𝐴^†𝐶+𝐷𝐵^†−𝐴^†𝐴𝐷𝐵^†+𝐼−𝐴^†𝐴𝑊𝐼−𝐵𝐵^†, (13) where 𝑊∈ℂ^𝑛×𝑝 is arbitrary.

By applying the singular value decomposition, similar to the proof of Theorem  1 in [24], the following lemma can be shown.

Lemma 13. Assume 𝐸,𝐹∈ℂ^𝑚×𝑛. Let the singular value decomposition of 𝐸 be 𝐸=𝑈𝑉Σ000^∗, (14) where 𝑈∈𝑈ℂ^𝑚×𝑚,𝑉∈𝑈ℂ^𝑛×𝑛,Σ=diag𝛼₁,…,𝛼_𝑟,𝛼_𝑖(𝐸).>0,𝑖=1,…,𝑟;𝑟=𝑟 (15) Partition 𝑉𝑋𝑉^∗=𝑋₁₁𝑋₁₂𝑋^∗₁₂𝑋₂₂,𝑈^∗𝐹𝐹𝑉=¹¹𝐹₁₂𝐹₂₁𝐹₂₂, (16) where 𝑋₁₁∈𝐻ℂ^𝑟×𝑟,𝑋₂₂∈𝐻ℂ^{(𝑛−𝑟)×(𝑛−𝑟)},𝐹₁₁∈ℂ^𝑟×𝑟,𝐹₂₂∈ℂ^{(𝑚−𝑟)×(𝑛−𝑟)}. (17) Then the matrix equation 𝐸𝑋=𝐹 (18) has Hermitian solutions if and only if 𝐸𝐸^†𝐹=𝐹,𝐸𝐹^∗=𝐹𝐸^∗,𝐹₂₁=0,𝐹₂₂=0, (19) in which case the Hermitian solution can be expressed as Σ𝑋=𝑉⁻¹𝐹₁₁Σ⁻¹𝐹₁₂𝐹^∗₁₂Σ⁻¹𝑋₂₂𝑉^∗, (20) where 𝑋₂₂∈𝐻ℂ^{(𝑛−𝑟)×(𝑛−𝑟)} is arbitrary.

By the similar way, the following lemma can also be verified.

Lemma 14. Assume 𝑀,𝑁∈ℂ^𝑚×𝑛. Let the singular value decomposition of 𝑀 be 𝑀=𝑈𝑉Π000^∗, (21) where 𝑈∈𝑈ℂ^𝑚×𝑚,𝑉∈𝑈ℂ^𝑛×𝑛,Π=diag𝛽₁,…,𝛽_𝑠,𝛽_𝑖(𝑀).>0,𝑖=1,…,𝑠;𝑠=𝑟 (22) Partition 𝑉𝑋𝑉^∗=𝑋₁₁𝑋₁₂−𝑋^∗₁₂𝑋₂₂,𝑈^∗𝑁𝑁𝑉=¹¹𝑁₁₂𝑁₂₁𝑁₂₂, (23) where 𝑋₁₁∈𝐴𝐻ℂ^𝑠×𝑠,𝑋₂₂∈𝐴𝐻ℂ^{(𝑛−𝑠)×(𝑛−𝑠)},𝑁₁₁∈ℂ^𝑠×𝑠,𝑁₂₂∈ℂ^{(𝑚−𝑠)×(𝑛−𝑠)}. (24) Then the matrix equation 𝑀𝑋=𝑁 (25) has an anti-Hermitian solution if and only if 𝑀𝑀^†𝑁=𝑁,𝑀𝑁^∗=−𝑁𝑀^∗,𝑁₂₁=0,𝑁₂₂=0, (26) in which case the anti-Hermitian solution can be expressed as Π𝑋=𝑉⁻¹𝑁₁₁Π⁻¹𝑁₁₂−𝑁^∗₁₂Π⁻¹𝑋₂₂𝑉^∗, (27) where 𝑋₂₂∈𝐴𝐻ℂ^{(𝑛−𝑠)×(𝑛−𝑠)} is arbitrary.

Lemma 15 (see [31]). Given 𝐴^′,𝐵^′∈ℂ^{𝑘×(𝑚+𝑞)}, 𝐶^′,𝐷^′∈ℂ^{𝑘×(𝑚+𝑞)}, suppose that the matrices 𝐴^′ and 𝐶^′, respectively, have the following singular value decompositions: 𝐴^′=𝑃₁𝑄Γ000^∗1,𝐶^′=𝑈₁𝑉Λ000^∗1, (28) where 𝑃₁=𝑃₁₁𝑃₁₂∈𝑈ℂ^𝑘×𝑘,𝑃₁₁∈ℂ^𝑘×𝑡1;𝑄₁=𝑄₁₁𝑄₁₂∈𝑈ℂ^{(𝑚+𝑞)×(𝑚+𝑞)},𝑄₁₁∈ℂ^{(𝑚+𝑞)×𝑡1};𝑈₁=𝑈₁₁𝑈₁₂∈𝑈ℂ^𝑘×𝑘,𝑈₁₁∈ℂ^𝑘×𝑡2;𝑉₁=𝑉₁₁𝑉₁₂∈𝑈ℂ^{(𝑚+𝑞)×(𝑚+𝑞)},𝑉₁₁∈ℂ^{(𝑚+𝑞)×𝑡2};Γ=diag𝛿₁,𝛿₂,…,𝛿_𝑡1,𝛿_𝑖>0,1≤𝑖≤𝑡₁;𝑡₁=𝑟𝐴^′;Λ=diag𝛾₁,𝛾₂,…,𝛾_𝑡2,𝛾_𝑖>0,1≤𝑖≤𝑡₂;𝑡₂=𝑟𝐶^′. (29) Then the solution set of the problem 𝑓𝑋=₁₂^Δ𝐴^′∗𝑋₁₂−𝐵^′∗2+𝑋₁₂𝐶^′−𝐷^′2=min (30) consists of matrices 𝑋₁₂∈ℂ^𝑘×𝑘 with the following form: 𝑋₁₂=𝑃₁𝜙∗𝑃^∗₁₁𝐷^′𝑉₁₁Λ+Γ𝑄^∗₁₁𝐵^′∗𝑈₁₂Γ⁻¹𝑄^∗₁₁𝐵^′∗𝑈₁₂𝑃^∗₁₂𝐷^′𝑉₁₁Λ⁻¹𝑋^′₂₂𝑈^∗₁, (31) where 𝜙=𝜙_𝑖𝑗,𝜙_𝑖𝑗=1𝛿²_𝑖+𝛾²_𝑗,1≤𝑖≤𝑡₁,1≤𝑗≤𝑡₂, (32) and 𝑋^′₂₂∈ℂ^{(𝑘−𝑡1)×(𝑘−𝑡2)} is arbitrary.

Lemma 16. Given 𝐸,𝐹∈ℂ^𝑚×𝑛, let the singular value decomposition of 𝐸, the partitions of 𝑉𝑋𝑉^∗ and 𝑈^∗𝐹𝑉 be, respectively, as in (14)–(16). Then the least squares Hermitian solution to the matrix equation (18) can be expressed as 𝑋=𝑉Φ∗Σ𝐹₁₁+𝐹^∗₁₁ΣΣ⁻¹𝐹₁₂𝐹^∗₁₂Σ⁻¹𝑋₂₂𝑉^∗, (33) where 1Φ=𝛼²_𝑖+𝛼²_𝑗,1≤𝑖,𝑗≤𝑟, (34) and 𝑋₂₂∈𝐻ℂ^{(𝑛−𝑟)×(𝑛−𝑟)} is arbitrary.

Proof. Combining (14)–(16) and the unitary invariance of the Frobenius norm, it is easy to obtain that ‖𝐸𝑋−𝐹‖²=𝑉Σ000^∗𝑉𝑋₁₁𝑋₁₂𝑋^∗₁₂𝑋₂₂−𝑈^∗𝐹𝑉²=Σ𝑋₁₁Σ𝑋₁₂−𝐹00¹¹𝐹₁₂𝐹₂₁𝐹₂₂²=Σ𝑋₁₁−𝐹₁₁²+Σ𝑋₁₂−𝐹₁₂²+𝐹₂₁²+𝐹₂₂². (35) Then ‖𝐸𝑋−𝐹‖² reaches its minimum if and only if Σ𝑋₁₁−𝐹₁₁²,Σ𝑋₁₂−𝐹₁₂² (36)(37) reach their minimum. For 𝑋₁₁=(𝑥_𝑖𝑗)∈𝐻ℂ^𝑟×𝑟, 𝐹₁₁=(𝑓_𝑖𝑗)∈ℂ^𝑟×𝑟, since 𝑥_𝑖𝑗=𝑥^∗_𝑖𝑗,1≤𝑖,𝑗≤𝑟, then Σ𝑋₁₁−𝐹₁₁²=^𝑟𝑟_𝑖=1𝑗=1𝛼_𝑖𝑥_𝑖𝑗−𝑓_𝑖𝑗²=^𝑟_{1≤𝑖,𝑗≤𝑟}𝛼²_𝑖+𝛼²_𝑗𝑥_𝑖𝑗²+2𝛼_𝑖𝑓_𝑖𝑗+𝛼_𝑗𝑓^∗_𝑖𝑗𝑥_𝑖𝑗+2𝑓_𝑖𝑗². (38) Hence, there exists a unique solution 𝑋₁₁=(𝑥_𝑖𝑗)∈𝐻ℂ^𝑟×𝑟 for (36) such that 𝑥_𝑖𝑗=𝛼_𝑖𝑓_𝑖𝑗+𝛼_𝑗𝑓^∗_𝑖𝑗𝛼²_𝑖+𝛼²_𝑗,1≤𝑖,𝑗≤𝑟. (39) That is, 𝑋₁₁=Φ∗Σ𝐹₁₁+𝐹^∗₁₁Σ, (40) where 1Φ=𝛼²_𝑖+𝛼²_𝑗,1≤𝑖,𝑗≤𝑟. (41) When 𝑋₁₂ can be expressed as 𝑋₁₂=Σ⁻¹𝐹₁₂, (42) (37) gets its minimum. Therefore, the least squares Hermitian solution to (18) can be described as (33).

By the similar way, the following result can be obtained.

Lemma 17. Given 𝑀,𝑁∈ℂ^𝑚×𝑛, let the singular value decomposition of 𝑀, the partitions of 𝑉𝑋𝑉^∗, and 𝑈^∗𝑁𝑉 be, respectively, as in (21)–(23). Then the least squares anti-Hermitian solution to the matrix equation 𝑀𝑋=𝑁 can be expressed as 𝑋=𝑉Ψ∗Π𝑁₁₁−𝑁^∗₁₁ΠΠ⁻¹𝑁₁₂−𝑁^∗₁₂Π⁻¹𝑋₂₂𝑉^∗, (43) where 1Ψ=𝛽²_𝑖+𝛽²_𝑗,1≤𝑖,𝑗≤𝑠, (44) and 𝑋₂₂∈𝐴𝐻ℂ^{(𝑛−𝑠)×(𝑛−𝑠)} is arbitrary.

Lemma 18 (see [20]). Given 𝐹∈ℂ^𝑚×𝑛, 𝐺∈ℂ^𝑝×𝑞, and 𝐿∈ℂ^𝑚×𝑞, then the matrix equation 𝐹𝑋𝐺=𝐿 has a solution if and only if 𝐹𝐹^†𝐿𝐺^†𝐺=𝐿, (45) in which case the general solution is 𝑋=𝐹^†𝐿𝐺^†+𝑌−𝐹^†𝐹𝑌𝐺𝐺^†, (46) where 𝑌∈ℂ^𝑛×𝑝 is arbitrary.

The following lemma is due to [25, 32] or [29, Lemma  5].

Lemma 19. Let 𝑀,𝑁∈ℂ^𝑚×𝑛. Then there exists a unique matrix 𝑊₁∈ℂ^𝑚×𝑛 such that 𝑊₁−𝑀²+𝑊₁−𝑁²=min_{𝑊∈ℂ𝑚×𝑛}‖𝑊−𝑀‖²+‖𝑊−𝑁‖², (47) where 𝑊₁=𝑀+𝑁2. (48)

3. The Solvability Conditions and the Expression of the Solution to Problem 5

In this section, our purpose is to derive the necessary and sufficient conditions of and the explicit expression of the solution to Problem 5 by using the results introduced in Section 2.

Theorem 20. Given 𝐴,𝐵∈ℂ^𝑚×2𝑘,𝐶,𝐷∈ℂ^2𝑘×𝑞, let the decomposition of 𝑋∈𝐻𝐻ℂ^2𝑘×2𝑘 be (7). Partition 𝐴𝑃=𝐴₁𝐴₂,𝐴₁∈ℂ^𝑚×𝑘,𝐴₂∈ℂ^𝑚×𝑘𝐵;𝐵𝑃=₁𝐵₂,𝐵₁∈ℂ^𝑚×𝑘,𝐵₂∈ℂ^𝑚×𝑘𝑃_∗;𝐶𝐶=¹𝐶₂,𝐶₁∈ℂ^𝑘×𝑞,𝐶₂∈ℂ^𝑘×𝑞𝑃_∗;𝐷𝐷=¹𝐷₂,𝐷₁∈ℂ^𝑘×𝑞,𝐷₂∈ℂ^𝑘×𝑞𝐴_′;=𝐴₁𝐶^∗₁,𝐵^′=𝐵₂𝐷^∗₂𝐶_′,=𝐴^∗₂𝐶₂,𝐷^′=𝐵^∗₁𝐷₁. (49)(50)(51)(52)(53)(54) Then Problem 5 has a solution 𝑋∈𝐻𝐻ℂ^2𝑘×2𝑘 if and only if 𝐴^′𝐴^′†𝐵^′=𝐵^′,𝐷^′𝐶^′†𝐶^′=𝐷^′,𝐴^′𝐷^′=𝐵^′𝐶^′, (55) in which case the Hermitian generalized Hamiltonian solution to Problem 5 can be expressed as 𝑋=𝑃0𝑋₁₂𝑋^∗₁₂0𝑃^∗, (56) where 𝑋₁₂=𝐴^′†𝐵^′+𝐷^′𝐶^′†−𝐴^′†𝐴^′𝐷^′𝐶^′†+𝐿_𝐴′𝑊𝑅_𝐶′ (57) and 𝑊∈ℂ^𝑘×𝑘 is arbitrary.

Proof. It follows from (7) and (49)–(52) that the system (3) can be transformed into the following system of matrix equations: 𝐴₁𝑋₁₂=𝐵₂,𝑋₁₂𝐴^∗₂=𝐵^∗₁,𝐶^∗₁𝑋₁₂=𝐷^∗₂,𝑋₁₂𝐶₂=𝐷₁. (58) Then, combining (53) and (54) yields that 𝐴^′𝑋₁₂=𝐵^′,𝑋₁₂𝐶^′=𝐷^′. (59) Thus, by Lemma 12, the system (59) has a solution 𝑋₁₂∈ℂ^𝑘×𝑘 if and only if all equalities in (55) hold, in which case the solution can be written as (57). So the solution to system (3) can be expressed as (56).

Remark 21. Let 𝐶 and 𝐷 vanish in Theorem 20. Partition 𝐴𝑃=𝐴₁𝐴₂,𝐴₁∈ℂ^𝑚×𝑘,𝐴₂∈ℂ^𝑚×𝑘;𝐵𝑃=𝐵₁𝐵₂,𝐵₁∈ℂ^𝑚×𝑘,𝐵₂∈ℂ^𝑚×𝑘. (60) Then the matrix equation 𝐴𝑋=𝐵 has Hermitian generalized Hamiltonian solutions if and only if 𝐴₁𝐴^†₁𝐵₂=𝐵₂,𝐴₂𝐴^†₂𝐵₁=𝐵₁,𝐴₁𝐵^∗₁=𝐵₂𝐴^∗₂, (61) in which case its solution can be described as 𝑋=𝑃0𝑋₁₂𝑋^∗₁₂0𝑃^∗, (62) where 𝑋₁₂=𝐴^†₁𝐵₂+𝐵^∗₁𝐴^†₂^∗−𝐴^†₁𝐴₁𝐵^∗₁𝐴^†₂^∗+𝐿_𝐴1𝑊𝐿_𝐴2 (63) and 𝑊∈ℂ^𝑘×𝑘 is arbitrary. It is clear that this result is different from Theorem  3.1 given in [1].

Similarly, by Lemmas 9 and 12, we can get the anti-Hermitian generalized anti-Hamiltonian solution to system (3).

Theorem 22. Given 𝐴,𝐵∈ℂ^𝑚×2𝑘, 𝐶,𝐷∈ℂ^2𝑘×𝑞, let the decomposition of 𝑋∈𝐴𝐻𝐴𝐻ℂ^2𝑘×2𝑘 be (8). 𝐴𝑃,𝐵𝑃,𝑃^∗𝐶, and 𝑃^∗𝐷, respectively, have the partitions as in (49)–(52). Put 𝐴𝐴=¹𝐶^∗₁,𝐵𝐵=²−𝐷^∗₂,𝐶=𝐴^∗₂𝐶₂,𝐷=−𝐵^∗₁𝐷₁. (64) Then Problem 5 has a solution 𝑋∈𝐴𝐻𝐴𝐻ℂ^2𝑘×2𝑘 if and only if 𝐴𝐴^†𝐵=𝐵,𝐷𝐶^†𝐶=𝐷,𝐴𝐷=𝐵𝐶, (65) in which case the anti-Hermitian generalized anti-Hamiltonian solution to Problem 5 can be expressed as 𝑋=𝑃0𝑋₁₂−𝑋^∗₁₂0𝑃^∗, (66) where 𝑋₁₂=𝐴^†𝐵+𝐷𝐶^†−𝐴^†𝐴𝐷𝐶^†𝐴𝐶+𝐿𝑍𝑅 (67) and 𝑍∈ℂ^𝑘×𝑘 is arbitrary.

Now, we investigate the Hermitian generalized anti-Hamiltonian solution to the system (3).

Theorem 23. Given 𝐴,𝐵∈ℂ^𝑚×2𝑘, 𝐶,𝐷∈ℂ^2𝑘×𝑞, let the decomposition of 𝑋∈𝐻𝐴𝐻ℂ^2𝑘×2𝑘 be (9). 𝐴𝑃,𝐵𝑃,𝑃^∗𝐶, and 𝑃^∗𝐷, respectively, have the partitions as in (49)–(52). Denote 𝐴𝐴=¹𝐶^∗₁,𝐵𝐵=¹𝐷^∗₁,𝐴𝐶=²𝐶^∗₂,𝐵𝐷=²𝐷^∗₂. (68)(69) Let the singular value decompositions of 𝐴 and 𝐶 be, respectively, 𝐴=𝑈𝑉Σ000^∗,𝐶=𝑄𝑅Π000^∗, (70)(71) where 𝑈∈𝑈ℂ^{(𝑚+𝑞)×𝑘},𝑉∈𝑈ℂ^𝑘×𝑘,Σ=diag𝛼₁,…,𝛼_𝑟,𝛼_𝑖>0,𝑖=1,…,𝑟;𝑟=𝑟𝐴,𝑄∈𝑈ℂ^{(𝑚+𝑞)×𝑘},𝑅∈𝑈ℂ^𝑘×𝑘,Π=diag𝛽₁,…,𝛽_𝑠,𝛽_𝑗>0,𝑗=1,…,𝑠;𝑠=𝑟𝐶. (72) Set 𝑉𝑋₁₁𝑉^∗=𝑋₁₁𝑋₁₂𝑋^∗₁₂𝑋₂₂𝑈_∗;𝐵𝑉=𝐵₁₁𝐵₁₂𝐵₂₁𝐵₂₂;𝑅𝑋₂₂𝑅^∗=𝑋₁₁𝑋₁₂𝑋^∗₁₂𝑋₂₂𝑄_∗;𝐷𝑅=𝐷₁₁𝐷₁₂𝐷₂₁𝐷₂₂, (73)(74)(75)(76) where 𝑋₁₁∈𝐻ℂ^𝑟×𝑟,𝑋₁₁∈𝐻ℂ^𝑠×𝑠,𝑋₂₂∈𝐻ℂ^{(𝑘−𝑟)×(𝑘−𝑟)},𝑋₂₂∈𝐻ℂ^{(𝑘−𝑠)×(𝑘−𝑠)},𝐵₁₁∈ℂ^𝑟×𝑟,𝐷₁₁∈ℂ^𝑠×𝑠,𝐵₂₂∈ℂ^{(𝑚+𝑞−𝑟)×(𝑘−𝑟)},𝐷₂₂∈ℂ^{(𝑚+𝑞−𝑠)×(𝑘−𝑠)}. (77) Then Problem 5 has a solution 𝑋∈𝐻𝐴𝐻ℂ^2𝑘×2𝑘 if and only if 𝐴𝐴^†𝐵=𝐵,𝐴𝐵^∗=𝐵𝐴^∗,𝐵₂₁=0,𝐵₂₂=0,𝐶𝐶^†𝐷=𝐷,𝐶𝐷^∗=𝐷𝐶^∗,𝐷₂₁=0,𝐷₂₂=0, (78)(79) in which case the Hermitian generalized anti-Hamiltonian solution to Problem 5 can be described as 𝑋𝑋=𝑃¹¹00𝑋₂₂𝑃^∗, (80) where 𝑋₁₁Σ=𝑉⁻¹𝐵₁₁Σ⁻¹𝐵₁₂𝐵₁₂^∗Σ⁻¹𝑋₂₂𝑉^∗𝑋₂₂,Π=𝑅⁻¹𝐷₁₁Π⁻¹𝐷₁₂𝐷₁₂^∗Π⁻¹𝑋₂₂𝑅^∗, (81)(82) and 𝑋₂₂∈𝐻ℂ^{(𝑘−𝑟)×(𝑘−𝑟)}, 𝑋₂₂∈𝐻ℂ^{(𝑘−𝑠)×(𝑘−𝑠)} are arbitrary.

Proof. It can be derived from (9), (49)–(52), and (68)-(69) that the system (3) is consistent if and only if the following two equations: 𝐴𝑋₁₁=𝐵,𝐶𝑋₂₂=𝐷, (83)(84) are solvable. By (70), (73), and (74), and then combining Lemma 13, we can obtain that there exists Hermitian solution 𝑋₁₁ such that (83) holds if and only if all equalities in (78) hold, in which case the solution can be written as (81). By the similar way, there exists Hermitian solution 𝑋₂₂ such that (84) holds if and only if all equalities in (79) hold, in which case the solution can be described as (82). Therefore, the Hermitian generalized anti-Hamiltonian solution to Problem 5 can be expressed as (80).

From Lemmas 11 and 14, it is not difficult to obtain the anti-Hermitian generalized Hamiltonian solution to Problem 5, which can be described as follows.

Theorem 24. Given 𝐴,𝐵∈ℂ^𝑚×2𝑘, 𝐶,𝐷∈ℂ^2𝑘×𝑞, let the decomposition of 𝑋∈𝐴𝐻𝐻ℂ^2𝑘×2𝑘 be (10). 𝐴𝑃,𝐵𝑃,𝑃^∗𝐶, and 𝑃^∗𝐷, respectively, have the partitions as in (49)–(52). Denote 𝐴𝐴=¹𝐶^∗₁,𝐵𝐵=¹−𝐷^∗₁,𝐴𝐶=²𝐶^∗₂,𝐵𝐷=²−𝐷^∗₂. (85) Let the singular value decompositions of 𝐴 and 𝐶 be, respectively, 𝐴=𝑈𝑉Σ000^∗,𝐶=𝑄𝑅Π000^∗, (86) where 𝑈∈𝑈ℂ^{(𝑚+𝑞)×𝑘},𝑉∈𝑈ℂ^𝑘×𝑘,Σ=diag𝛼₁,…,𝛼_𝑟,𝛼_𝑖𝐴,>0,𝑖=1,…,𝑟;𝑟=𝑟𝑄∈𝑈ℂ^{(𝑚+𝑞)×𝑘},𝑅∈𝑈ℂ^𝑘×𝑘,Π=diag𝛽₁,…,𝛽_𝑠,𝛽_𝑗𝐶.>0,𝑗=1,…,𝑠;𝑠=𝑟 (87) Set 𝑉𝑋₁₁𝑉^∗=𝑋₁₁𝑋₁₂−𝑋^∗₁₂𝑋₂₂;𝑈^∗𝐵𝐵𝑉=¹¹𝐵₁₂𝐵₂₁𝐵₂₂;𝑅𝑋₂₂𝑅^∗=𝑋₁₁𝑋₁₂−𝑋^∗₁₂𝑋₂₂;𝑄^∗𝐷𝐷𝑅=¹¹𝐷₁₂𝐷₂₁𝐷₂₂, (88) where 𝑋₁₁∈𝐻ℂ^𝑟×𝑟,𝑋₁₁∈𝐻ℂ^𝑠×𝑠,𝑋₂₂∈𝐻ℂ^{(𝑘−𝑟)×(𝑘−𝑟)},𝑋₂₂∈𝐻ℂ^{(𝑘−𝑠)×(𝑘−𝑠)},𝐵₁₁∈ℂ^𝑟×𝑟,𝐷₁₁∈ℂ^𝑠×𝑠,𝐵₂₂∈ℂ^{(𝑚+𝑞−𝑟)×(𝑘−𝑟)},𝐷₂₂∈ℂ^{(𝑚+𝑞−𝑠)×(𝑘−𝑠)}. (89) Then Problem 5 has a solution 𝑋∈𝐴𝐻𝐻ℂ^2𝑘×2𝑘 if and only if 𝐴𝐴^†𝐵=𝐵,𝐴𝐵^∗=−𝐵𝐴^∗,𝐵₂₁=0,𝐵₂₂𝐶𝐶=0,_†𝐷=𝐷,𝐶𝐷^∗=−𝐷𝐶^∗,𝐷₂₁=0,𝐷₂₂=0, (90) in which case the anti-Hermitian generalized Hamiltonian solution to Problem 5 can be described as 𝑋𝑋=𝑃¹¹00𝑋₂₂𝑃^∗, (91) where 𝑋₁₁Σ=𝑉⁻¹𝐵₁₁Σ⁻¹𝐵₁₂−𝐵^∗₁₂Σ⁻¹𝑋₂₂𝑉^∗,𝑋₂₂Π=𝑅⁻¹𝐷₁₁Π⁻¹𝐷₁₂−𝐷^∗₁₂Π⁻¹𝑋₂₂𝑅^∗, (92) and 𝑋₂₂∈𝐴𝐻ℂ^{(𝑘−𝑟)×(𝑘−𝑟)}, 𝑋₂₂∈𝐴𝐻ℂ^{(𝑘−𝑠)×(𝑘−𝑠)} are arbitrary.

4. The Expression of the Unique Solution to Problem 6

In this section, our aim is to derive the optimal approximation solution to Problem 6.

Theorem 25. Given 𝑋∈ℂ^2𝑘×2𝑘, under the hypotheses of Theorem 20, let 𝑃^∗𝑋𝑋𝑃=¹¹𝑋₁₂𝑋₂₁𝑋₂₂,𝑋₁₁∈ℂ^𝑘×𝑘,𝑋₂₂∈ℂ^𝑘×𝑘. (93) If Problem 5 has Hermitian generalized Hamiltonian solutions, then Problem 6 has a unique solution 𝑋∈𝐻𝐻ℂ^2𝑘×2𝑘 if and only if 𝐿_𝐴′𝑋₁₂+𝑋₂₁^∗2−𝑋₀𝑅_𝐶′=𝑋₁₂+𝑋₂₁^∗2−𝑋₀, (94) in which case the unique solution 𝑋 can be expressed as 0𝑋=𝑃𝑋₀𝑋₀^∗0𝑃^∗, (95) where 𝑋₀=𝑋₁₂+𝑋₂₁^∗2,𝑋₀=𝐴^′†𝐵^′+𝐷^′𝐶^′†−𝐴^′†𝐴^′𝐷^′𝐶^′†. (96)

Proof. When the Hermitian generalized Hamiltonian solution set 𝐾 of Problem 5 is nonempty, it is not difficult to verify that 𝐾 is a closed convex set. Then by [33], Problem 6 has a unique solution 𝑋∈𝐻𝐻ℂ^2𝑘×2𝑘. From Theorem 20, for any 𝑋∈𝐾, 𝑋 can be expressed as 𝑋=𝑃0𝑋₀𝑋^∗₀0𝑃^∗+𝑃0𝐿_𝐴′𝑊𝑅_𝐶′𝑅_𝐶′𝑊^∗𝐿_𝐴′0𝑃^∗, (97) where 𝑋₀=𝐴^′†𝐵^′+𝐷^′𝐶^′†−𝐴^′†𝐴^′𝐷^′𝐶^′† (98) and 𝑊∈ℂ^𝑘×𝑘 is arbitrary. Then it follows from the equalities in (93) and (97) and the unitary invariance of the Frobenius norm that 𝑋−𝑋²=𝑃^∗𝑋𝑃−𝑃^∗𝑋𝑃²=𝑋₁₁𝑋₁₂−𝑋₀−𝐿_𝐴′𝑊𝑅_𝐶′𝑋₂₁−𝑋^∗₀−𝑅_𝐶′𝑊^∗𝐿_𝐴′𝑋₂₂²=𝑋₁₁²+𝑋₂₂²+𝑋₁₂−𝑋₀−𝐿_𝐴′𝑊𝑅_𝐶′²+𝑋₂₁−𝑋^∗₀−𝑅_𝐶′𝑊^∗𝐿_𝐴′²=𝑋₁₁²+𝑋₂₂²+𝐿_𝐴′𝑊𝑅_𝐶′−−𝑋₀+𝑋₁₂²+𝐿_𝐴′𝑊𝑅_𝐶′−−𝑋₀+𝑋^∗₂₁². (99) Thus, Problem 6 has a unique solution 𝑋∈𝐻𝐻ℂ^2𝑘×2𝑘 if and only if there exists 𝑊 such that 𝐿_𝐴′𝑊𝑅_𝐶′−−𝑋₀+𝑋₁₂²+𝐿_𝐴′𝑊𝑅_𝐶′−−𝑋₀+𝑋₂₁^∗2 (100) reaches its minimum. Therefore, by Lemma 19, (100) arrives at its minimum if and only if there exists 𝑊 such that the matrix equation 𝐿_𝐴′𝑊𝑅_𝐶′=−𝑋₀+𝑋₁₂−𝑋₀+𝑋₂₁^∗2=𝑋₁₂+𝑋₂₁^∗2−𝑋₀ (101) holds, which, by Lemma 18, has a solution if and only if (94) holds, in which case the solution can be expressed as 𝑊=𝐿_𝐴′𝑋₁₂+𝑋₂₁^∗2−𝑋₀𝑅_𝐶′+𝑍−𝐿_𝐴′𝑍𝑅_𝐶′, (102) where 𝑍∈ℂ^𝑘×𝑘 is arbitrary. Inserting (102) into (97), and then combining (94) yields (95).

Analogously, the following theorem can be shown.

Theorem 26. Given 𝑋∈ℂ^2𝑘×2𝑘, under the hypotheses of Theorem 22, let 𝑃^∗𝑋𝑋𝑃=¹¹𝑋₁₂𝑋₂₁𝑋₂₂,𝑋₁₁∈ℂ^𝑘×𝑘,𝑋₂₂∈ℂ^𝑘×𝑘. (103) If Problem 5 has anti-Hermitian generalized anti-Hamiltonian solutions, then Problem 6 has a unique solution 𝑋∈𝐴𝐻𝐴𝐻ℂ^2𝑘×2𝑘 if and only if 𝐿𝐴𝑋₁₂−𝑋₂₁^∗2−𝑋₀𝑅𝐶=𝑋₁₂−𝑋₂₁^∗2−𝑋₀, (104) in which case the unique solution 𝑋 can be expressed as 0𝑋=𝑃𝑋₀−𝑋₀^∗0𝑃^∗, (105) where 𝑋₀=𝑋₁₂−𝑋₂₁^∗2,𝑋₀=𝐴^†𝐵+𝐷𝐶^†−𝐴^†𝐴𝐷𝐶^†. (106)

Now, we give the unique Hermitian generalized anti-Hamiltonian solution to Problem 6.

Theorem 27. Given 𝑋∈ℂ^2𝑘×2𝑘, under the hypotheses of Theorem 23, let 𝑃^∗𝑋+𝑋^∗2𝑋𝑃=^′11𝑋^′₁₂𝑋^′₁₂^∗𝑋^′₂₂,𝑋^′₁₁∈𝐻ℂ^𝑘×𝑘,𝑋^′₂₂∈𝐻ℂ^𝑘×𝑘;𝑉^∗𝑋^′₁₁𝑋𝑉=^∘11𝑋^∘₁₂𝑋^∘₁₂^∗𝑋^∘₂₂,𝑋^∘₁₁∈𝐻ℂ^𝑟×𝑟,𝑋^∘₂₂∈𝐻ℂ^{(𝑘−𝑟)×(𝑘−𝑟)};𝑅^∗𝑋^′₂₂𝑋𝑅=^′11𝑋^′′₁₂𝑋^′′₁₂^∗𝑋^′′₂₂,𝑋^′′₁₁∈𝐻ℂ^𝑠×𝑠,𝑋^′′₂₂∈𝐻ℂ^{(𝑘−𝑠)×(𝑘−𝑠)}. (107)(108)(109) If Problem 5 has Hermitian generalized anti-Hamiltonian solutions, then the unique solution 𝑋∈𝐻𝐴𝐻ℂ^2𝑘×2𝑘 to Problem 6 can be expressed as 𝑋𝑋=𝑃^∘1100𝑋^∘₂₂𝑃^∗, (110) where 𝑋^∘₁₁Σ=𝑉⁻¹𝐵₁₁Σ⁻¹𝐵₁₂𝐵₁₂^∗Σ⁻¹𝑋^∘₂₂𝑉^∗𝑋_∘22,Π=𝑅⁻¹𝐷₁₁Π⁻¹𝐷₁₂𝐷₁₂^∗Π⁻¹𝑋^′′₂₂𝑅^∗. (111)(112)

Proof. When the Hermitian generalized anti-Hamiltonian solution set 𝐾 of Problem 5 is nonempty, it is easy to prove that 𝐾 is a closed convex set. Then, Problem 6 has a unique solution 𝑋∈𝐻𝐴𝐻ℂ^2𝑘×2𝑘 by the aid of [33]. For any 𝑋∈𝐾, due to Theorem 23, 𝑋 can be expressed as 𝑋𝑋=𝑃¹¹00𝑋₂₂𝑃^∗, (113) where 𝑋₁₁ and 𝑋₂₂ have the expressions as in (81) and (82). Combining the equalities in (80)–(82) and (107) and the unitary invariance of the Frobenius norm yields that 𝑋−𝑋²=𝑋𝑋−𝑋+^∗2²+𝑋−𝑋^∗2²=𝑃𝑋₁₁00𝑋₂₂𝑃^∗−𝑋+𝑋^∗2²+𝑋−𝑋^∗2²=𝑋₁₁00𝑋₂₂−𝑋^′₁₁𝑋^′₁₂𝑋^′₁₂^∗𝑋^′₂₂²+𝑋−𝑋^∗2²=𝑋₁₁−𝑋^′₁₁²+𝑋₂₂−𝑋^′₂₂²+2𝑋^′₁₂²+𝑋−𝑋^∗2². (114) So, min_{𝑋∈∈𝐻𝐴𝐻ℂ2𝑘×2𝑘}𝑋−𝑋²holds⟺min_{𝑋11∈𝐻ℂ𝑘×𝑘}𝑋₁₁−𝑋^′₁₁²holdsandmin_{𝑋22∈𝐻ℂ𝑘×𝑘}𝑋₂₂−𝑋^′₂₂²holds. (115) By (81), (108), and the unitary invariance of the Frobenius norm, we obtain 𝑋₁₁−𝑋^′₁₁²=Σ⁻¹𝐵₁₁Σ⁻¹𝐵₁₂𝐵₁₂^∗Σ⁻¹𝑋₂₂−𝑋^∘₁₁𝑋^∘₁₂𝑋^∘₁₂^∗𝑋^∘₂₂²=Σ⁻¹𝐵₁₁−𝑋^∘₁₁²+𝑋₂₂−𝑋^∘₂₂²+2Σ⁻¹𝐵₁₂−𝑋^∘₁₂². (116) Then min_{𝑋11∈𝐻ℂ𝑘×𝑘}𝑋₁₁−𝑋^′₁₁²holds⟺min_{𝑋22∈𝐻ℂ(𝑘−𝑟)×(𝑘−𝑟)}𝑋₂₂−𝑋^∘₂₂²holds. (117) Therefore, when 𝑋₂₂ can be expressed as 𝑋₂₂=𝑋^∘₂₂, (118)min_{𝑋11∈𝐻ℂ}^𝑘×𝑘‖𝑋₁₁−𝑋^′₁₁‖² holds. Then combining (81) yields (111). Similarly, we can derive the expression in (112) by (82) and (109). Thus, (110) is the unique solution to Problem 6.

By the method used in Theorem 27, the following theorem can also be shown.

Theorem 28. Given 𝑋∈ℂ^2𝑘×2𝑘, under the hypotheses of Theorem 24, let 𝑃^∗𝑋−𝑋^∗2𝑋𝑃=^′11𝑋^′₁₂−𝑋^′₁₂^∗𝑋^′₂₂,𝑋^′₁₁∈𝐴𝐻ℂ^𝑘×𝑘,𝑋^′₂₂∈𝐴𝐻ℂ^𝑘×𝑘;𝑉^∗𝑋^′₁₁𝑋𝑉=^∘11𝑋^∘₁₂−𝑋^∘₁₂^∗𝑋^∘₂₂,𝑋^∘₁₁∈𝐴𝐻ℂ^𝑟×𝑟,𝑋^∘₂₂∈𝐴𝐻ℂ^{(𝑘−𝑟)×(𝑘−𝑟)};𝑅^∗𝑋^′₂₂𝑋𝑅=^′′11𝑋^′′₁₂−𝑋^′′₁₂^∗𝑋^′′₂₂,𝑋^′′₁₁∈𝐴𝐻ℂ^𝑠×𝑠,𝑋^′′₂₂∈𝐴𝐻ℂ^{(𝑘−𝑠)×(𝑘−𝑠)}. (119) If Problem 5 has anti-Hermitian generalized Hamiltonian solutions, then the unique solution 𝑋∈𝐴𝐻𝐻ℂ^2𝑘×2𝑘 to Problem 6 can be expressed as 𝑋𝑋=𝑃^∘1100𝑋^∘₂₂𝑃^∗, (120) where 𝑋^∘₁₁Σ=𝑉⁻¹𝐵₁₁Σ⁻¹𝐵₁₂−𝐵^∗₁₂Σ⁻¹𝑋^∘₂₂𝑉^∗,𝑋^∘₂₂Π=𝑅⁻¹𝐷₁₁Π⁻¹𝐷₁₂−𝐷^∗₁₂Π⁻¹𝑋^′′₂₂𝑅^∗. (121)

5. The Expression of the Solution to Problem 7

If the solvability conditions of linear matrix equations are not satisfied, the least squares solution is usually considered. So, in this section, the solution to Problem 7 is constructed.

Theorem 29. Given 𝐴,𝐵∈ℂ^𝑚×2𝑘, 𝐶,𝐷∈ℂ^2𝑘×𝑞, let the decomposition of 𝑋∈𝐻𝐻ℂ^2𝑘×2𝑘 be (7). 𝐴𝑃,𝐵𝑃,𝑃^∗𝐶,𝑃^∗𝐷,𝐶^′, and 𝐷^′, respectively, have the partitions as in (49)–(52) and (54). Denote 𝐴^′=𝐴^∗₁𝐶₁,𝐵^′=𝐵^∗₂𝐷₂. (122) Let the singular value decompositions of 𝐴^′ and 𝐶^′ be as given in (28). Then the least squares Hermitian generalized Hamiltonian solution to Problem 7 can be described as (7), where 𝑋₁₂ has the expression as in (31).

Proof. Combining (7), (49)–(52), (54), (122), and the unitary invariance of the Frobenius norm yields that ‖𝐴𝑋−𝐵‖²+‖𝑋𝐶−𝐷‖²=𝐴^′∗𝑋₁₂−𝐵^′∗2+𝑋₁₂𝐶^′−𝐷^′2. (123) Therefore, by Lemma 15, if 𝑋₁₂ has the expression as in (31), then (123) reaches its minimum. Then, substituting (31) into (7), we obtain the least squares Hermitian generalized Hamiltonian solution to Problem 7.

Corollary 30. Given 𝐴,𝐵∈ℂ^𝑚×2𝑘, 𝐶,𝐷∈ℂ^2𝑘×𝑞, under the conditions of Theorem 29, the least squares Hermitian generalized Hamiltonian solution with minimum norm to Problem 7 can be described as (7), where 𝑋₁₂ has the expression as in (31) with 𝑋^′₂₂=0.

By the same way, we can also derive the least squares anti-Hermitian generalized anti-Hamiltonian solution to Problem 7.

Theorem 31. Given 𝐴,𝐵∈ℂ^𝑚×2𝑘, 𝐶,𝐷∈ℂ^2𝑘×𝑞, let the decomposition of 𝑋∈𝐴𝐻𝐴𝐻ℂ^2𝑘×2𝑘 be (8). 𝐴𝑃,𝐵𝑃,𝑃^∗𝐶, and 𝑃^∗𝐷, respectively, have the partitions as in (49)–(52). Denote 𝐴^′=𝐴^∗₁𝐶₁,𝐵^′=𝐵^∗₂−𝐷₂,𝐶^′=𝐴^∗₂𝐶₂,𝐷^′=−𝐵^∗₁𝐷₁. (124) Let the singular value decompositions of 𝐴^′ and 𝐶^′ be as in (28). Then the least squares anti-Hermitian generalized anti-Hamiltonian solution to Problem 7 can be described as (8), where 𝑋₁₂ has the expression as in (31).

Corollary 32. Given 𝐴,𝐵∈ℂ^𝑚×2𝑘, 𝐶,𝐷∈ℂ^2𝑘×𝑞, under the conditions of Theorem 31, the least squares anti-Hermitian generalized anti-Hamiltonian solution with minimum norm to Problem 7 can be described as (8), where 𝑋₁₂ has the expression as in (31) with 𝑋^′₂₂=0.

At present, we give the least squares Hermitian generalized anti-Hamiltonian solution to Problem 7.

Theorem 33. Assume 𝐴,𝐵∈ℂ^𝑚×2𝑘, 𝐶,𝐷∈ℂ^2𝑘×𝑞. Let the decomposition of 𝑋∈𝐻𝐴𝐻ℂ^2𝑘×2𝑘 be (9). 𝐴𝑃,𝐵𝑃,𝑃^∗𝐶, 𝑃^∗𝐷, 𝐴,𝐵,𝐶, and 𝐷, respectively, have the partitions as in (49)–(52), (68), and (69). Let the singular value decompositions of 𝐴 and 𝐶 be, respectively, (70) and (71), 𝑉𝑋₁₁𝑉^∗,𝑈^∗𝐵𝑉, 𝑅𝑋₂₂𝑅^∗,and𝑄^∗𝐷𝑅 have the partitions as in (73)–(76). Then the least squares Hermitian generalized anti-Hamiltonian solution to Problem 7 can be expressed as (9) with 𝑋₁₁Φ=𝑉¹∗Σ𝐵₁₁+𝐵^∗₁₁ΣΣ⁻¹𝐵₁₂𝐵^∗₁₂Σ⁻¹𝑋₂₂𝑉^∗𝑋₂₂,Φ=𝑅²∗Π𝐷₁₁+𝐷^∗₁₁ΠΠ⁻¹𝐷₁₂𝐷^∗₁₂Π⁻¹𝑋₂₂𝑅^∗, (125)(126) where Φ₁=1𝛼²_𝑖+𝛼²_𝑗Φ,1≤𝑖,𝑗≤𝑟;₂=1𝛽²_𝑖+𝛽²_𝑗,1≤𝑖,𝑗≤𝑠, (127) and 𝑋₂₂∈𝐻ℂ^{(𝑘−𝑟)×(𝑘−𝑟)}, 𝑋₂₂∈𝐻ℂ^{(𝑘−𝑠)×(𝑘−𝑠)} are arbitrary.

Proof. It follows from (9), (49)–(52), (68), (69), and the unitary invariance of the Frobenius norm that ‖𝐴𝑋−𝐵‖²+‖𝑋𝐶−𝐷‖²=𝐴𝑋₁₁−𝐵²+𝐶𝑋₂₂−𝐷². (128) Then ‖𝐴𝑋−𝐵‖²+‖𝑋𝐶−𝐷‖² (129) gains its minimum value if and only if min=𝐴𝑋₁₁−𝐵²holds,min=𝐶𝑋₂₂−𝐷²holds. (130)(131) So, by (68), (70), (73), and (74) and then combining Lemma 16, we get that if 𝑋₁₁ has the expression as in (125), then (130) holds. Similarly, if 𝑋₂₂ has the expression as in (126), then (131) holds. Thus, the least squares Hermitian generalized anti-Hamiltonian solution to Problem 7 can be expressed as (9), where 𝑋₁₁ and 𝑋₂₂ have the expressions as in (125) and (126).

Corollary 34. Given 𝐴,𝐵∈ℂ^𝑚×2𝑘, 𝐶,𝐷∈ℂ^2𝑘×𝑞, under the conditions of Theorem 33, the least squares Hermitian generalized anti-Hamiltonian solution with minimum norm to Problem 7 can be expressed as (9) with 𝑋₁₁ and 𝑋₂₂ having the expressions as in (125) and (126), where 𝑋₂₂=0,𝑋₂₂=0.

At last, on the basis of Lemma 17, we can obtain the least squares anti-Hermitian generalized Hamiltonian solution to Problem 7, the proof of which is analogous to the proof of Theorem 33.

Theorem 35. Given 𝐴,𝐵∈ℂ^𝑚×2𝑘, 𝐶,𝐷∈ℂ^2𝑘×𝑞, let the decomposition of 𝑋∈𝐴𝐻𝐻ℂ^2𝑘×2𝑘 be (10). 𝐴𝑃,𝐵𝑃,𝑃^∗𝐶,𝑃^∗𝐷,𝐴,𝐵,𝐶, and 𝐷, respectively, have the partitions as in (49)–(52), (85). Assume that the singular value decompositions of 𝐴 and 𝐶 are, respectively, expressed as in (86) and 𝑉𝑋₁₁𝑉^∗,𝑈^∗𝐵𝑉, 𝑅𝑋₂₂𝑅^∗,𝑄^∗𝐷𝑅 have the partitions as in (88). Then the least squares anti-Hermitian generalized Hamiltonian solution to Problem 7 can be expressed as (10) with 𝑋₁₁Ψ=𝑉¹∗Σ𝐵₁₁−𝐵^∗₁₁ΣΣ⁻¹𝐵₁₂−𝐵^∗₁₂Σ⁻¹𝑋₂₂𝑉^∗,𝑋₂₂Ψ=𝑅²∗Π𝐷₁₁−𝐷^∗₁₁ΠΠ⁻¹𝐷₁₂−𝐷^∗₁₂Π⁻¹𝑋₂₂𝑅^∗, (132) where Ψ₁=1𝛼²_𝑖+𝛼²_𝑗Ψ,1≤𝑖,𝑗≤𝑟;₂=1𝛽²_𝑖+𝛽²_𝑗,1≤𝑖,𝑗≤𝑠, (133) and 𝑋₂₂∈𝐴𝐻ℂ^{(𝑘−𝑟)×(𝑘−𝑟)}, 𝑋₂₂∈𝐴𝐻ℂ^{(𝑘−𝑠)×(𝑘−𝑠)} are arbitrary.

Corollary 36. Given 𝐴,𝐵∈ℂ^𝑚×2𝑘, 𝐶,𝐷∈ℂ^2𝑘×𝑞, under the conditions of Theorem 35, the least squares anti-Hermitian generalized Hamiltonian solution with minimum norm to Problem 7 can be expressed as (10) with 𝑋₁₁ and 𝑋₂₂ having the expressions as in (132), where 𝑋₂₂=0,𝑋₂₂=0.

6. Algorithms and Numerical Examples

In this section, algorithms are given to compute the solution to Problem 7, and meanwhile some numerical examples are presented to show that the algorithms provided are feasible. Note that all the tests are performed by MATLAB 7.6.

An algorithm is firstly presented to compute the least squares Hermitian generalized Hamiltonian solution to Problem 7.

Algorithm 37. Step  1. Input 𝐴,𝐵,𝐶,𝐷,𝐽.
Step  2. Compute the eigenvalue decomposition of 𝐽 according to (6).
Step  3. Compute 𝐴𝑃,𝐵𝑃,𝑃^∗𝐶,𝑃^∗𝐷 according to (49)–(52).
Step  4. Compute 𝐴^′,𝐵^′,𝐶^′,𝐷^′ according to (53) and (54). If the conditions in (55) hold, then compute the Hermitian generalized Hamiltonian solution to Problem 5 according to (56) and (57). Otherwise, turn to Step  5.
Step  5. Compute 𝐴^′,𝐵^′,𝐶^′,𝐷^′ according to (53) and (122).
Step  6. Compute the singular value decompositions of 𝐴^′ and 𝐶^′ according to (28).
Step  7. Compute 𝑋₁₂ according to (31).
Step  8. Compute 𝑋 according to (7), and output 𝑋.

Example 38. Given ,,,,,𝐴=3+6𝑖2+𝑖7−2𝑖8+3𝑖2−3𝑖5−4𝑖1+4𝑖9+3𝑖𝐵=2−4𝑖3+2𝑖5+𝑖4+𝑖6+𝑖2−5𝑖1+6𝑖5+3𝑖𝐶=4+7𝑖10+3𝑖7+𝑖8+7𝑖3+9𝑖1−6𝑖2−5𝑖5+6𝑖2+7𝑖23𝑖3+7𝑖𝐷=7+3𝑖52𝑖5+2𝑖2−3𝑖6−𝑖3+𝑖9−𝑖44−2𝑖5+2𝑖1+4𝑖𝐽=000100−100100−1000 (134)

it can be easily verified that the conditions in (55) are not satisfied. Then, according to Algorithm 37, the least squares Hermitian generalized Hamiltonian solution 𝑋 to Problem 7 can be expressed as 𝑋=,0.0865−𝑖0.0141−0.0359𝑖0.0317−0.0312𝑖0.1174+0.0824𝑖0.0141+0.0359𝑖−0.0148−0.0201−0.0682𝑖0.1189+0.0954𝑖0.0317+0.0302𝑖−0.0201+0.0682𝑖−0.07240.1114+0.0359𝑖0.1173−0.0824𝑖0.1189−0.0954𝑖0.1114−0.0359𝑖0.0006min_{𝑋∈𝐻𝐻ℂ2𝑘×2𝑘}𝑋−𝑋^∗=0.0000,min_{𝑋∈𝐻𝐻ℂ2𝑘×2𝑘}𝑋^∗−𝐽𝑋𝐽=0.6000. (135)

Remark 39. (1) There exists a unique least squares Hermitian generalized Hamiltonian solution to Problem 7 if and only if both 𝐴^′ and 𝐶^′ in Theorem 29 have full row ranks. Example 38 just illustrates it.
(2) Similarly, the algorithm about computing the least squares anti-Hermitian generalized anti-Hamiltonian solution to Problem 7 can be shown. We omit it here.
Now, we provide another algorithm to compute the least squares Hermitian generalized anti-Hamiltonian solution to Problem 7.

Algorithm 40. Step  1. Input 𝐴,𝐵,𝐶,𝐷,𝐽.
Step  2. Compute the eigenvalue decomposition of 𝐽 according to (6).
Step  3. Compute 𝐴𝑃,𝐵𝑃,𝑃^∗𝐶,𝑃^∗𝐷 according to (49)–(52).
Step  4. Compute 𝐴,𝐵,𝐶,𝐷 according to (68) and (69).
Step  5. Compute the singular value decompositions of 𝐴 and 𝐶 according to (70)-(71).
Step  6. Compute the partitions of 𝑈^∗𝐵𝑉,𝑄^∗𝐷𝑅 according to (74) and (76). If the conditions in (78) and (79) are all satisfied, then compute the Hermitian generalized anti-Hamiltonian solution to Problem 5 according to (80)–(82). Otherwise, turn to Step  7.
Step  7. Compute 𝑋₁₁ and 𝑋₂₂ according to (125) and (126).
Step  8. Compute 𝑋 according to (9), and output 𝑋.

Example 41. Let 𝐴,𝐵,𝐶,𝐷,𝐽 be as given in Example 38.
It is not difficult to prove that the conditions in (78) and (79) do not hold. So, according to Algorithm 40, the least squares Hermitian generalized anti-Hamiltonian solution to Problem 7 can be written as 𝑋=,0.36710.1579−0.1804𝑖0.1822+0.1400𝑖0.0179−0.0012𝑖0.1579+0.1804𝑖0.23240.0179+0.1149𝑖−0.0662−0.1400𝑖0.1822−0.1400𝑖0.0179−0.1149𝑖0.21430.1221−0.0849𝑖0.0179+0.0012𝑖−0.0662+0.1400𝑖0.1221+0.0849𝑖0.5764min_{𝑋∈𝐻𝐴𝐻ℂ2𝑘×2𝑘}𝑋−𝑋^∗=0.0000,min_{𝑋∈𝐻𝐴𝐻ℂ2𝑘×2𝑘}𝑋^∗+𝐽𝑋𝐽=0.8309. (136)

Remark 42. (1) There exists a unique least squares Hermitian generalized anti-Hamiltonian solution to Problem 7 if and only if both 𝐴 and 𝐶 in Theorem 33 have full column ranks. Example 41 is just the case.
(2) Similarly, the algorithm about computing the least squares anti-Hermitian generalized Hamiltonian solution to Problem 7 can be obtained. We also omit it here.

7. Conclusions

In the previous sections, using the decomposition of the (anti-)Hermitian generalized (anti-)Hamiltonian matrices, the necessary and sufficient conditions for the existence of and the expression for the solution to Problem 5 have been firstly derived, respectively. Then the solutions to Problems 6 and 7 have been individually given. Finally, algorithms have been given to compute the least squares Hermitian generalized Hamiltonian solution and the least squares Hermitian generalized anti-Hamiltonian solution to Problem 7, and the corresponding examples have also been presented to show that the algorithms are reasonable.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the Grants from the National Natural Science Foundation of China (11171205), the National Natural Science Foundation of China (11301330), the Key Project of Scientific Research Innovation Foundation of Shanghai Municipal Education Commission (13ZZ080), the Natural Science Foundation of Shanghai (11ZR1412500), the Youth Funds of Natural Science Foundation of Hebei province (A2012403013), and the Natural Science Foundation of Hebei province (A2012205028).