An Iterative Algorithm for the Least Squares Generalized Reflexive Solutions of the Matrix Equations 𝐴𝑋𝐵=𝐸, 𝐶𝑋𝐷=𝐹

Abstract

The generalized coupled Sylvester systems play a fundamental role in wide applications in several areas, such as stability theory, control theory, perturbation analysis, and some other fields of pure and applied mathematics. The iterative method is an important way to solve the generalized coupled Sylvester systems. In this paper, an iterative algorithm is constructed to solve the minimum Frobenius norm residual problem: ‖^𝐴𝑋𝐵_𝐶𝑋𝐷−^𝐸_𝐹‖= min over generalized reflexive matrix 𝑋. For any initial generalized reflexive matrix 𝑋₁, by the iterative algorithm, the generalized reflexive solution 𝑋^∗ can be obtained within finite iterative steps in the absence of round-off errors, and the unique least-norm generalized reflexive solution 𝑋^∗ can also be derived when an appropriate initial iterative matrix is chosen. Furthermore, the unique optimal approximate solution 𝑋 to a given matrix 𝑋₀ in Frobenius norm can be derived by finding the least-norm generalized reflexive solution 𝑋^∗ of a new corresponding minimum Frobenius norm residual problem: min‖^𝐴𝑋_𝐵𝐶𝑋_𝐷−𝐸𝐹‖ with 𝐸=𝐸−𝐴𝑋₀𝐵, 𝐹=𝐹−𝐶𝑋₀𝐷. Finally, several numerical examples are given to illustrate that our iterative algorithm is effective.

1. Introduction

A matrix 𝑃∈ℛ^𝑛×𝑛 is said to be a generalized reflection matrix if 𝑃 satisfies that 𝑃^𝑇=𝑃,𝑃²=𝐼. Let 𝑃∈ℛ^𝑚×𝑚 and 𝑄∈ℛ^𝑛×𝑛 be two generalized reflection matrices. A matrix 𝐴∈ℛ^𝑚×𝑛 is called generalized reflexive (or generalized anti-reflexive) with respect to the matrix pair (𝑃,𝑄) if 𝑃𝐴𝑄=𝐴(or𝑃𝐴𝑄=−𝐴). The set of all m-by-n generalized reflexive matrices with respect to matrix pair (𝑃,𝑄) is denoted by ℛ_𝑟^𝑚×𝑛(𝑃,𝑄). The generalized reflexive and anti-reflexive matrices have many special properties and usefulness in engineering and scientific computations [1–3].

In this paper, we will consider the minimum Frobenius norm residual problem and its optimal approximation problem as follows.

Problem 1. For given matrices 𝐴∈ℛ^𝑝×𝑚,𝐵∈ℛ^𝑛×𝑞,𝐶∈ℛ^𝑠×𝑚, 𝐷∈ℛ^𝑛×𝑡,𝐸∈ℛ^𝑝×𝑞,𝐹∈ℛ^𝑠×𝑡, find matrix 𝑋∈ℛ_𝑟^𝑚×𝑛(𝑃,𝑄) such that −𝐸𝐹𝐴𝑋𝐵𝐶𝑋𝐷=min. (1.1)

Problem 2. Let 𝑆_𝐸 denote the set of the generalized reflexive solutions of Problem 1. For a given matrix 𝑋₀∈ℛ_𝑟^𝑚×𝑛(𝑃,𝑄), find 𝑋∈𝑆_𝐸 such that 𝑋−𝑋₀=min_{𝑋∈𝑆𝐸}𝑋−𝑋₀. (1.2)

Problem 1 plays a fundamental role in wide applications in several areas, such as Pole assignment, measurement feedback, and matrix programming problem.Liao and Lei [4]presented some examples to show a motivation for studying Problem 1. Problem 2 arises frequently in experimental design. Here the matrix 𝑋₀ may be a matrix obtained from experiments, but it may not satisfy the structural requirement (generalized reflexive) and/or spectral requirement (the solution of Problem 1). The best estimate 𝑋 is the matrix that satisfies both requirements and is the best approximation of 𝐴 in the Frobenius norm.

Least-squares-based iterative algorithms are very important in system identification, parameter estimation, and signal processing, including the recursive least squares (RLS) and iterative least squares (ILS) methods for solving the solutions of some matrix equations, for example, the Lyapunov matrix equation, Sylvester matrix equations, and coupled matrix equations as well. Some related contributions in solving matrix equations and parameter identification/estimation should be mentioned in this paper. For example, novel gradient-based iterative (GI) method [5–9] and least-squares-based iterative methods [5, 9, 10] with highly computational efficiencies for solving (coupled) matrix equations are presented and have good stability performances, based on the hierarchical identification principle [11–13] which regards the unknown matrix as the system parameter matrix to be identified.

The explicit and numerical solutions of matrix equation pair 𝐴𝑋𝐵=𝐸,𝐶𝑋𝐷=𝐹 have been addressed in a large body of the literature. Peng et al. [14] presented iterative methods to obtain the symmetric solutions of the matrix equation pair. Sheng and Chen [15] presented a finite iterative method when the matrix equation pair is consistent. Liao and Lei [4] presented an analytical expression of the least squares solution and an algorithm for the matrix equation pair with the minimum norm. Peng et al. [16] presented an efficient algorithm for the least squares reflexive solution. Dehghan and Hajarian [17] presented an iterative algorithm for solving a pair of the matrix equations over generalized centrosymmetric matrices. Cai and Chen [18] presented an iterative algorithm for the least squares bisymmetric solutions of the matrix equations. By applying the hierarchical identification principle,Kılıçman and Zhour [19]developed an iterative algorithm for obtaining the weighted least squares solution. Dehghan and Hajarian [20] constructed an iterative algorithm to solve the generalized coupled Sylvester matrix equations (𝐴𝑋𝐵+𝐶𝑌𝐷,𝐸𝑋𝐹+𝐺𝑌𝐻)=(𝑀,𝑁) over generalized bisymmetric matrices. Wu et al. [21, 22] gave the finite iterative solutions to coupled Sylvester-conjugate matrix equations. Wu et al. [23] gave the finite iterative solutions to a class of complex matrix equations with conjugate and transpose of the unknowns. Jonsson and Kågström [24, 25] proposed recursive block algorithms for solving the coupled Sylvester matrix equations and the generalized Sylvester and Lyapunov Matrix equations. Very recently, Huang et al. [26] presented a finite iterative algorithms for the one-sided and generalized coupled Sylvester matrix equations over generalized reflexive solutions. Yin et al. [27] presented a finite iterative algorithms for the two-sided and generalized coupled Sylvester matrix equations over reflexive solutions. For more studies on the matrix equations, we refer to [1–4, 16, 17, 28–40]. However, the problem of finding the least squares generalized reflexive solution of the matrix equation pair has not been solved.

The following notations are also used in this paper. Let ℛ^𝑚×𝑛 denote the set of all 𝑚×𝑛 real matrices. We denote by the superscript 𝑇 the transpose of a matrix. In matrix space ℛ^𝑚×𝑛, define inner product as ⟨𝐴,𝐵⟩=trace(𝐵^𝑇𝐴) for all 𝐴,𝐵∈ℛ^𝑚×𝑛, and ‖𝐴‖ represents the Frobenius norm of 𝐴. 𝑅(𝐴) represents the column space of 𝐴. vec(⋅) represents the vector operator, that is, vec(𝐴)=(𝐚^𝑇₁,𝐚^𝑇₂,…,𝐚^𝑇_𝑛)^𝑇∈ℛ^𝑚𝑛 for the matrix 𝐴=(𝐚₁,𝐚₂,…,𝐚_𝑛)∈ℛ^𝑚×𝑛,𝐚_𝑖∈ℛ^𝑚,𝑖=1,2,…,𝑛. 𝐴⊗𝐵 stands for the Kronecker product of matrices 𝐴 and 𝐵.

This paper is organized as follows. In Section 2, we will solve Problem 1 by constructing an iterative algorithm, that is, for an arbitrary initial matrix 𝑋₁∈ℛ_𝑟^𝑚×𝑛(𝑃,𝑄), we can obtain a solution 𝑋^∗∈ℛ_𝑟^𝑚×𝑛(𝑃,𝑄) of Problem 1 within finite iterative steps in the absence of round-off errors. The convergence of the algorithm is also proved. Let 𝑋₁=𝐴^𝑇𝐻𝐵^𝑇+𝐶^𝑇𝐻𝐷^𝑇+𝑃𝐴^𝑇𝐻𝐵^𝑇𝑄+𝑃𝐶^𝑇𝐻𝐷^𝑇𝑄, where 𝐻∈ℛ^𝑝×𝑞,𝐻∈ℛ^𝑠×𝑡 are arbitrary matrices, or more especially, let 𝑋₁=𝟎∈ℛ_𝑟^𝑚×𝑛(𝑃,𝑄);we can obtain the unique least-norm solution 𝑋^∗ of Problem 1. Then in Section 3, we give the optimal approximate solution of Problem 2 by finding the least-norm generalized reflexive solution of a corresponding new minimum Frobenius norm residual problem. In Section 4, several numerical examples are given to illustrate the application of our iterative algorithm.

2. Solution of Problem 1

In this section, we firstly introduce some definitions, lemmas, and theorems which are required for solving Problem 1. Then we present an iterative algorithm to obtain the solution of Problem 1. We also prove that it is convergent. The following definitions and lemmas come from [41], which are needed for our derivation.

Definition 2.1. A set of matrices 𝑆∈ℛ^𝑚×𝑛 is said to be convex if for 𝑋₁,𝑋₂∈𝑆 and 𝛼∈(0,1),𝛼𝑋₁+(1−𝛼)𝑋₂∈𝑆. Let ℛ_𝑐 denote a convex subset of ℛ^𝑚×𝑛.

Definition 2.2. A matrix function 𝑓∶ℛ_𝑐→ℛ is said to be convex if 𝑓𝛼𝑋₁+(1−𝛼)𝑋₂≤𝛼𝑓𝑋₁+(1−𝛼)𝑓𝑋₂ (2.1) for 𝑋₁,𝑋₂∈ℛ_𝑐 and 𝛼∈(0,1).

Definition 2.3. Let 𝑓∶ℛ_𝑐→ℛ be a continuous and differentiable function. The gradient of 𝑓 is defined as ∇𝑓(𝑋)=(𝜕𝑓(𝑋)/𝜕𝑥_𝑖𝑗).

Lemma 2.4. Let 𝑓∶ℛ_𝑐→ℛ be a continuous and differentiable function. Then 𝑓 is convex on ℛ_𝑐 if and only if 𝑓(𝑌)≥𝑓(𝑋)+∇𝑓(𝑋),𝑌−𝑋 (2.2) for all 𝑋,𝑌∈ℛ_𝑐.

Lemma 2.5. Let 𝑓∶ℛ_𝑐→ℛ be a continuous and differentiable function, and there exists 𝑋^∗ in the interior of ℛ_𝑐 such that 𝑓(𝑋^∗)=min_{𝑋∈ℛ𝑐}𝑓(𝑋), then ∇𝑓(𝑋^∗)=0.

Note that the set ℛ_𝑟^𝑚×𝑛(𝑃,𝑄) is unbounded, open, and convex. Denote 𝐹(𝑋)=−𝐸𝐹𝐴𝑋𝐵𝐶𝑋𝐷², (2.3) then 𝐹(𝑋) is a continuous, differentiable, and convex function on ℛ_𝑟^𝑚×𝑛(𝑃,𝑄). Hence, by applying Lemmas 2.4 and 2.5, we obtain the following lemma.

Lemma 2.6. Let 𝐹(𝑋) be defined by (2.3), then there exists 𝑋^∗∈ℛ_𝑟^𝑚×𝑛(𝑃,𝑄) if and only if 𝐹(𝑋^∗)=min_{𝑋∈ℛ𝑟𝑚×𝑛(𝑃,𝑄)}𝐹(𝑋), then ∇𝐹(𝑋^∗)=0.

From the Taylor series expansion, we have 𝐹(𝑋+𝜖𝑌)=𝐹(𝑋)+𝜖⟨∇𝐹(𝑋),𝑌⟩+𝑜(𝜖),∀𝑋,𝑌∈ℛ_𝑟^𝑚×𝑛(𝑃,𝑄),𝜖∈ℛ. (2.4) On the other hand, by the basic properties of Frobenius norm and the matrix inner product, we get the expression𝐹()=−𝐸𝐹𝑋+𝜖𝑌𝐴(𝑋+𝜖𝑌)𝐵𝐶(𝑋+𝜖𝑌)𝐷²=⟨⟩+⟨⟩=⟨⟩+⟨⟩⟨⟩⟨⟩𝐴𝑋𝐵−𝐸+𝜖𝐴𝑌𝐵,𝐴𝑋𝐵−𝐸+𝜖𝐴𝑌𝐵𝐶𝑋𝐷−𝐹+𝜖𝐶𝑌𝐷,𝐶𝑋𝐷−𝐹+𝜖𝐶𝑌𝐷𝐴𝑋𝐵−𝐸,𝐴𝑋𝐵−𝐸𝐶𝑋𝐷−𝐹,𝐶𝑋𝐷−𝐹+2𝜖𝐴𝑋𝐵−𝐸,𝐴𝑌𝐵+2𝜖𝐶𝑋𝐷−𝐹,𝐶𝑌𝐷+𝜖²⟩+⟨(𝑋)⟨⟩⟨⟩(⟨𝐴𝑌𝐵,𝐴𝑌𝐵𝐶𝑌𝐷,𝐶𝑌𝐷⟩)=𝐹+2𝜖𝐴𝑋𝐵−𝐸,𝐴𝑌𝐵+2𝜖𝐶𝑋𝐷−𝐹,𝐶𝑌𝐷+𝜖²(⟨𝐴𝑌𝐵,𝐴𝑌𝐵⟩+⟨𝐶𝑌𝐷,𝐶𝑌𝐷⟩). (2.5) Note that 2𝜖⟨𝐴𝑋𝐵−𝐸,𝐴𝑌𝐵⟩=2𝜖𝐴^𝑇𝐴𝑋𝐵𝐵^𝑇−𝐴^𝑇𝐸𝐵^𝑇𝐴,𝑌=2𝜖^𝑇𝐴𝑋𝐵𝐵^𝑇−𝐴^𝑇𝐸𝐵^𝑇+𝑃𝐴^𝑇𝐴𝑋𝐵𝐵^𝑇−𝐴^𝑇𝐸𝐵^𝑇𝑄2𝐴,𝑌+2𝜖^𝑇𝐴𝑋𝐵𝐵^𝑇−𝐴^𝑇𝐸𝐵^𝑇−𝑃𝐴^𝑇𝐴𝑋𝐵𝐵^𝑇−𝐴^𝑇𝐸𝐵^𝑇𝑄2𝐴,𝑌=𝜖^𝑇𝐴𝑋𝐵𝐵^𝑇−𝐴^𝑇𝐸𝐵^𝑇+𝑃𝐴^𝑇𝐴𝑋𝐵𝐵^𝑇𝑄−𝑃𝐴^𝑇𝐸𝐵^𝑇𝐴𝑄,𝑌+𝜖^𝑇𝐴𝑋𝐵𝐵^𝑇−𝐴^𝑇𝐸𝐵^𝑇−𝑃𝐴^𝑇𝐴𝑋𝐵𝐵^𝑇−𝐴^𝑇𝐸𝐵^𝑇𝐴𝑄,𝑌=𝜖^𝑇𝐴𝑋𝐵𝐵^𝑇−𝐴^𝑇𝐸𝐵^𝑇+𝑃𝐴^𝑇𝐴𝑋𝐵𝐵^𝑇𝑄−𝑃𝐴^𝑇𝐸𝐵^𝑇,⟨⟩𝐶𝑄,𝑌2𝜀𝐶𝑋𝐷−𝐹,𝐶𝑌𝐷=2𝜖^𝑇𝐶𝑋𝐷𝐷^𝑇−𝐶^𝑇𝐹𝐷^𝑇𝐶,𝑌=2𝜖^𝑇𝐶𝑋𝐷𝐷^𝑇−𝐶^𝑇𝐹𝐷^𝑇+𝑃𝐶^𝑇𝐶𝑋𝐷𝐷^𝑇−𝐶^𝑇𝐹𝐷^𝑇𝑄2𝐶,𝑌+2𝜖^𝑇𝐶𝑋𝐷𝐷^𝑇−𝐶^𝑇𝐹𝐷^𝑇−𝑃𝐶^𝑇𝐶𝑋𝐷𝐷^𝑇−𝐶^𝑇𝐹𝐷^𝑇𝑄2𝐶,𝑌=𝜖^𝑇𝐶𝑋𝐷𝐷^𝑇−𝐶^𝑇𝐹𝐷^𝑇+𝑃𝐶^𝑇𝐶𝑋𝐷𝐷^𝑇𝑄−𝑃𝐶^𝑇𝐹𝐷^𝑇𝐶𝑄,𝑌+𝜖^𝑇𝐶𝑋𝐷𝐷^𝑇−𝐶^𝑇𝐹𝐷^𝑇−𝑃𝐶^𝑇𝐶𝑋𝐷𝐷^𝑇−𝐶^𝑇𝐹𝐷^𝑇𝐶𝑄,𝑌=𝜖^𝑇𝐶𝑋𝐷𝐷^𝑇−𝐶^𝑇𝐹𝐷^𝑇+𝑃𝐶^𝑇𝐶𝑋𝐷𝐷^𝑇𝑄−𝑃𝐶^𝑇𝐹𝐷^𝑇𝑄,𝑌. (2.6) Thus, we have 𝐹(𝑋+𝜖𝑌)=𝐹(𝑋)+𝜖𝐴^𝑇𝐴𝑋𝐵𝐵^𝑇−𝐴^𝑇𝐸𝐵^𝑇+𝑃𝐴^𝑇𝐴𝑋𝐵𝐵^𝑇𝑄−𝑃𝐴^𝑇𝐸𝐵^𝑇𝑄+𝐶^𝑇𝐶𝑋𝐷𝐷^𝑇−𝐶^𝑇𝐹𝐷^𝑇+𝑃𝐶^𝑇𝐶𝑋𝐷𝐷^𝑇𝑄−𝑃𝐶^𝑇𝐹𝐷^𝑇𝑄,𝑌+𝜖²(⟨𝐴𝑌𝐵,𝐴𝑌𝐵⟩+⟨𝐶𝑌𝐷,𝐶𝑌𝐷⟩). (2.7) By comparing (2.4) with (2.7), we have ∇𝐹(𝑋)=𝐴^𝑇𝐴𝑋𝐵𝐵^𝑇−𝐴^𝑇𝐸𝐵^𝑇+𝑃𝐴^𝑇𝐴𝑋𝐵𝐵^𝑇𝑄−𝑃𝐴^𝑇𝐸𝐵^𝑇𝑄+𝐶^𝑇𝐶𝑋𝐷𝐷^𝑇−𝐶^𝑇𝐹𝐷^𝑇+𝑃𝐶^𝑇𝐶𝑋𝐷𝐷^𝑇𝑄−𝑃𝐶^𝑇𝐹𝐷^𝑇𝑄. (2.8)

According to Lemma 2.6 and (2.8), we obtain the following theorem.

Theorem 2.7. A matrix 𝑋^∗∈ℛ_𝑟^𝑚×𝑛(𝑃,𝑄) is a solution of Problem 1 if and only if ∇𝐹(𝑋^∗)=0.

For the convenience of discussion, we adopt the following notations: 𝑀(𝑋)=𝐴^𝑇𝐴𝑋𝐵𝐵^𝑇+𝐶^𝑇𝐶𝑋𝐷𝐷^𝑇+𝑃𝐴^𝑇𝐴𝑋𝐵𝐵^𝑇𝑄+𝑃𝐶^𝑇𝐶𝑋𝐷𝐷^𝑇𝑄,𝑁=𝐴^𝑇𝐸𝐵^𝑇+𝐶^𝑇𝐹𝐷^𝑇+𝑃𝐴^𝑇𝐸𝐵^𝑇𝑄+𝑃𝐶^𝑇𝐹𝐷^𝑇𝐺(𝑋)(𝑋)(𝑋),𝑃𝑄,=−∇𝐹=𝑁−𝑀_𝑘=𝐺𝑋_𝑘. (2.9) The following algorithm is constructed to solve Problems 1 and 2.

Algorithm 2.8. Step 1. Input matrices 𝐴∈ℛ^𝑝×𝑚,𝐵∈ℛ^𝑛×𝑞,𝐶∈ℛ^𝑠×𝑚, 𝐷∈ℛ^𝑛×𝑡,𝐸∈ℛ^𝑝×𝑞,𝐹∈ℛ^𝑠×𝑡, and two generalized reflection matrix 𝑃∈ℛ^𝑚×𝑚,𝑄∈ℛ^𝑛×𝑛;Step 2. Choose an arbitrary matrix 𝑋₁∈ℛ_𝑟^𝑚×𝑛(𝑃,𝑄). Compute 𝑃₁=𝑁−𝑀𝑋₁,𝑄₁=𝑀𝑃₁,𝑘∶=1. (2.10)Step 3. If 𝑃₁=0, then stop. Else go to Step 4.Step 4. Compute 𝑋_𝑘+1=𝑋_𝑘+𝑃_𝑘²⟨𝑄_𝑘,𝑀𝑃_𝑘⟩𝑄_𝑘,𝑃_𝑘+1=𝑃_𝑘−𝑃_𝑘²𝑄_𝑘,𝑀𝑃_𝑘𝑀𝑄_𝑘,𝑄_𝑘+1=𝑃_𝑘+1−𝑃_𝑘+1,𝑀𝑄_𝑘𝑄_𝑘,𝑀𝑄_𝑘𝑄_𝑘. (2.11)Step 5. If 𝑃_𝑘+1=0, then stop. Else, let 𝑘∶=𝑘+1, and go to Step 4.

Remark 2.9. Obviously, it can be seen that 𝑃_𝑖∈ℛ_𝑟^𝑚×𝑛(𝑃,𝑄),𝑄_𝑖∈ℛ_𝑟^𝑚×𝑛(𝑃,𝑄), and 𝑋_𝑖∈ℛ_𝑟^𝑚×𝑛(𝑃,𝑄), where 𝑖=1,2,….

Lemma 2.10. Suppose that 𝑋∈𝑅_𝑟^𝑚×𝑛(𝑃,𝑄),𝑌∈𝑅_𝑟^𝑚×𝑛(𝑃,𝑄), then ⟨𝑀(𝑋),𝑌⟩=⟨𝑋,𝑀(𝑌)⟩. (2.12)

Proof. One has⟨𝑀(𝑋)⟩=𝐴,𝑌_𝑇𝐴𝑋𝐵𝐵^𝑇+𝐶^𝑇𝐶𝑋𝐷𝐷^𝑇+𝑃𝐴^𝑇𝐴𝑋𝐵𝐵^𝑇𝑄+𝑃𝐶^𝑇𝐶𝑋𝐷𝐷^𝑇=𝐴𝑄,𝑌_𝑇𝐴𝑋𝐵𝐵^𝑇,𝑌+𝐶^𝑇𝐶𝑋𝐷𝐷^𝑇,𝑌+𝑃𝐴^𝑇𝐴𝑋𝐵𝐵^𝑇𝑄,𝑌+𝑃𝐶^𝑇𝐶𝑋𝐷𝐷^𝑇=𝑄,𝑌𝑋,𝐴^𝑇𝐴𝑌𝐵𝐵^𝑇+𝑋,𝐶^𝑇𝐶𝑌𝐷𝐷^𝑇+𝑋,𝐴^𝑇𝐴𝑃𝑌𝑄𝐵𝐵^𝑇+𝑋,𝐶^𝑇𝐶𝑃𝑌𝑄𝐷𝐷^𝑇=𝑋,𝐴^𝑇𝐴𝑌𝐵𝐵^𝑇+𝑋,𝐶^𝑇𝐶𝑌𝐷𝐷^𝑇+𝑃𝑋𝑄,𝐴^𝑇𝐴𝑃𝑌𝑄𝐵𝐵^𝑇+𝑃𝑋𝑄,𝐶^𝑇𝐶𝑃𝑌𝑄𝐷𝐷^𝑇=𝑋,𝐴^𝑇𝐴𝑌𝐵𝐵^𝑇+𝑋,𝐶^𝑇𝐶𝑌𝐷𝐷^𝑇+𝑋,𝑃𝐴^𝑇𝐴𝑌𝐵𝐵^𝑇𝑄+𝑋,𝑃𝐶^𝑇𝐶𝑌𝐷𝐷^𝑇𝑄=𝑋,𝐴^𝑇𝐴𝑌𝐵𝐵^𝑇+𝐶^𝑇𝐶𝑌𝐷𝐷^𝑇+𝑃𝐴^𝑇𝐴𝑌𝐵𝐵^𝑇𝑄+𝑃𝐶^𝑇𝐶𝑌𝐷𝐷^𝑇𝑄=⟨𝑋,𝑀(𝑌)⟩. (2.13) This completes the proof.

Lemma 2.11. For the sequences {𝑃_𝑖} and {𝑄_𝑖} generated by Algorithm 2.8, if there exists a positive number 𝑘 such that 𝑃_𝑖≠0 for all 𝑖=1,2,…,𝑘, then 𝑃_𝑖,𝑃_𝑗=0,𝑄_𝑖,𝑀𝑄_𝑗=0𝑖,𝑗=1,2,…,𝑘,𝑖≠𝑗. (2.14)

Proof. Since ⟨𝐴,𝐵⟩=⟨𝐵,𝐴⟩ holds for all matrices 𝐴 and 𝐵 in ℛ^𝑚×𝑛, we only need prove that ⟨𝑃_𝑖,𝑃_𝑗⟩=0,⟨𝑄_𝑖,𝑀(𝑄_𝑗)⟩=0 for all 1≤𝑖<𝑗≤𝑘. We prove the conclusion by induction and two steps are required.
Step 1. We will show that𝑃_𝑖,𝑃_𝑖+1=0,𝑄_𝑖,𝑀𝑄_𝑖+1=0,𝑖=1,2,…,𝑘−1. (2.15)
To prove this conclusion, we also use induction.
For 𝑖=1, by Algorithm 2.8 and Lemma 2.10, we have that𝑃₁,𝑃₂=𝑃₂,𝑃₁=𝑃₁−𝑃₁²𝑄₁,𝑀𝑃₁𝑀𝑄₁,𝑃₁=𝑃₁²−𝑃₁²𝑄₁,𝑀𝑃₁𝑀𝑄₁,𝑃₁𝑄=0,₁,𝑀𝑄₂=𝑄₂,𝑀𝑄₁=𝑃₂−𝑃₂,𝑀𝑄₁𝑄₁,𝑀𝑄₁𝑄₁,𝑀𝑄₁=𝑃₂,𝑀𝑄₁−𝑃₂,𝑀𝑄₁𝑄₁,𝑀𝑄₁𝑄₁,𝑀𝑄₁=0. (2.16)
Assume (2.15) holds for 𝑖=𝑠(1<𝑠<𝑘). For 𝑖=𝑠+1, by Lemma 2.10, we have⟨𝑃_𝑠+1,𝑃_𝑠+2⟩=𝑃_𝑠+2,𝑃_𝑠+1=𝑃_𝑠+1−𝑃_𝑠+1²𝑄_𝑠+1,𝑀𝑃_𝑠+1𝑀𝑄_𝑠+1,𝑃_𝑠+1=𝑃_𝑠+1²−𝑃_𝑠+1²𝑄_𝑠+1,𝑀𝑃_𝑠+1𝑀𝑄_𝑠+1,𝑃_𝑠+1=𝑃_𝑠+1²−𝑃_𝑠+1²𝑄_𝑠+1,𝑀𝑃_𝑠+1𝑄_𝑠+1,𝑀𝑃_𝑠+1𝑄=0,_𝑠+1,𝑀𝑄_𝑠+2=𝑄_𝑠+2,𝑀𝑄_𝑠+1=𝑃_𝑠+2−𝑃_𝑠+2,𝑀𝑄_𝑠+1𝑄_𝑠+1,𝑀𝑄_𝑠+1𝑄_𝑠+1,𝑀𝑄_𝑠+1=𝑃_𝑠+2,𝑀𝑄_𝑠+1−𝑃_𝑠+2,𝑀𝑄_𝑠+1𝑄_𝑠+1,𝑀𝑄_𝑠+1𝑄_𝑠+1,𝑀𝑄_𝑠+1=0. (2.17) Hence, (2.15) holds for 𝑖=𝑠+1. Therefore, (2.15) holds by the principle of induction.Step 2. Assume that ⟨𝑃_𝑖,𝑃_𝑠⟩=0,⟨𝑄_𝑖,𝑀(𝑄_𝑠)⟩=0,𝑖=1,2,…,𝑠−1, then we show that 𝑃_𝑖,𝑃_𝑠+1=0,𝑄_𝑖,𝑀𝑄_𝑠+1=0,𝑖=1,2,…,𝑠. (2.18) In fact, by Algorithm 2.8 we have 𝑃_𝑖,𝑃_𝑠+1=𝑃_𝑖,𝑃_𝑠−𝑃_𝑠²𝑄_𝑠,𝑀𝑃_𝑠𝑀𝑄_𝑠=⟨𝑃_𝑖,𝑃_𝑠⟩−‖𝑃_𝑠‖²⟨𝑄_𝑠,𝑀𝑃_𝑠⟩⟨𝑃_𝑖,𝑀𝑄_𝑠⟩𝑃=−^𝑠2𝑄_𝑠,𝑀𝑃_𝑠𝑄_𝑖+𝑃_𝑖,𝑀𝑄_𝑖−1𝑄_𝑖−1,𝑀𝑄_𝑖−1𝑄_𝑖−1,𝑀𝑄_𝑠𝑃=−^𝑠2𝑄_𝑠,𝑀𝑃_𝑠⟨𝑄_𝑖,𝑀𝑄_𝑖−1+𝑃_𝑠²𝑄_𝑠,𝑀𝑃_𝑠𝑃_𝑖,𝑀𝑄_𝑖−1𝑄_𝑖−1,𝑀𝑄_𝑖−1𝑄_𝑖−1,𝑀𝑄_𝑠𝑄=0,_𝑖,𝑀𝑄_𝑠+1=𝑀𝑄_𝑖,𝑄_𝑠+1=𝑀𝑄_𝑖,𝑃_𝑠+1−𝑃_𝑠+1,𝑀𝑄_𝑠𝑄_𝑠,𝑀𝑄_𝑠𝑄_𝑠=𝑀𝑄_𝑖,𝑃_𝑠+1−𝑃_𝑠+1,𝑀𝑄_𝑠𝑄_𝑠,𝑀𝑄_𝑠𝑀𝑄_𝑖,𝑄_𝑠=𝑄_𝑖,𝑀𝑃_𝑖𝑃_𝑖²𝑃_𝑖−𝑃_𝑖+1,𝑃_𝑠+1𝑄=−^𝑖,𝑀𝑃_𝑖𝑃_𝑖²𝑃_𝑖+1,𝑃_𝑠+1=0. (2.19) By the principle of induction, (2.14) is implied in Steps 1 and 2. This completes the proof.

Lemma 2.12. Assume that 𝑋^∗ is an arbitrary solution of Problem 1, then 𝑋^∗−𝑋_𝑘,𝑀𝑄_𝑘𝑃=2^𝑘2𝑃₁²𝐴𝑃₁𝐵²+𝐶𝑃₁𝐷²,𝑘=1,2,…, (2.20) where the sequences {𝑋_𝑘},{𝑃_𝑘}, and {𝑄_𝑘} are generated by Algorithm 2.8.

Proof. First, by Algorithm 2.8, it is easy to verify that −𝑃_𝑘+1,𝑀𝑄_𝑘𝑄_𝑘,𝑀𝑄_𝑘=𝑃_𝑘+1²𝑃_𝑘². (2.21) Thus 𝑋^∗−𝑋_𝑘,𝑀𝑄_𝑘=𝑀𝑋^∗−𝑋_𝑘,𝑄_𝑘=𝑃_𝑘,𝑄_𝑘=𝑃_𝑘−1−𝑃_𝑘−1²𝑄_𝑘−1,𝑀𝑃_𝑘−1𝑀𝑄_𝑘−1,𝑄_𝑘=𝑃_𝑘−1,𝑄_𝑘=⋯=𝑃₁,𝑄_𝑘=𝑃₁,𝑃_𝑘−𝑃_𝑘,𝑀𝑄_𝑘−1𝑄_𝑘−1,𝑀𝑄_𝑘−1𝑄_𝑘−1𝑃=−^𝑘,𝑀𝑄_𝑘−1𝑄_𝑘−1,𝑀𝑄_𝑘−1𝑃₁,𝑄_𝑘−1−𝑃=⋯=^𝑘,𝑀𝑄_𝑘−1𝑄_𝑘−1,𝑀𝑄_𝑘−1⋯−𝑃₂,𝑀𝑄₁𝑄₁,𝑀𝑄₁𝑃₁,𝑄₁=𝑃_𝑘²𝑃_𝑘−1²⋯𝑃₂²𝑃₁²𝑃₁,𝑀𝑃₁𝑃=2^𝑘2𝑃₁²𝐴𝑃₁𝐵²+𝐶𝑃₁𝐷². (2.22) This complete the proof.

Remark 2.13. Lemma 2.12 implies that if 𝑃_𝑖≠0, then 𝑀(𝑄_𝑖)≠0, thus 𝑄_𝑖≠0(𝑖=1,2,…).

Theorem 2.14. For an arbitrary initial matrix 𝑋₁∈ℛ_𝑟^𝑚×𝑛(𝑃,𝑄), a solution of Problem 1 can be obtained with finite iteration steps in the absence of round-off errors.

Proof. If 𝑃_𝑖≠0,𝑖=1,2,…,𝑚𝑛, by Lemma 2.12 we have 𝑄_𝑖≠0,𝑖=1,2,…,𝑚𝑛, then we can compute 𝑋_𝑚𝑛+1,𝑃_𝑚𝑛+1 by Algorithm 2.8.
By Lemma 2.11, we have 𝑃_𝑖,𝑃_𝑚𝑛+1𝑃=0,𝑖=1,2,…,𝑚𝑛,_𝑖,𝑃_𝑗=0,𝑖,𝑗=1,2,…,𝑚𝑛,𝑖≠𝑗. (2.23)
It can be seen that the set of 𝑃₁,𝑃₂,…,𝑃_𝑚𝑛 is an orthogonal basis of the matrix space ℛ_𝑟^𝑚×𝑛(𝑃,𝑄), which implies that 𝑃_𝑚𝑛+1=0, that is, 𝑋_𝑚𝑛+1 is a solution of Problem 1. This completes the proof.

To show the least-norm generalized reflexive solution of Problem 1, we first introduce the following result.

Lemma 2.15 (see [16, Lemma 2.7]). Suppose that the minimum residual problem ‖𝑀𝑦−𝑏‖=min has a solution 𝑦^∗∈𝑅(𝑀^𝑇), then 𝑦^∗ is the unique least Frobenius norm solution of the minimum residual problem.

By Lemma 2.15, the following result can be obtained.

Theorem 2.16. If one chooses the initial iterative matrix 𝑋₁=𝐴^𝑇𝐻𝐵^𝑇+𝐶^𝑇𝐻𝐷^𝑇+𝑃𝐴^𝑇𝐻𝐵^𝑇𝑄+𝑃𝐶^𝑇𝐻𝐷^𝑇𝑄, where 𝐻∈ℛ^𝑝×𝑞,𝐻∈ℛ^𝑠×𝑡 are arbitrary matrices, especially, let 𝑋₁=𝟎∈ℛ_𝑟^𝑚×𝑛, one can obtain the unique least-norm generalized reflexive solution of Problem 1 within finite iteration steps in the absence of round-off errors by using Algorithm 2.8.

Proof. By Algorithm 2.8 and Theorem 2.14, if we let 𝑋₁=𝐴^𝑇𝐻𝐵^𝑇+𝐶^𝑇𝐻𝐷^𝑇+𝑃𝐴^𝑇𝐻𝐵^𝑇𝑄+𝑃𝐶^𝑇𝐻𝐷^𝑇𝑄, where 𝐻∈ℛ^𝑝×𝑞,𝐻∈ℛ^𝑠×𝑡 are arbitrary matrices, we can obtain the solution 𝑋^∗ of Problem 1 within finite iteration steps in the absence of round-off errors, the solution 𝑋^∗ can be represented that 𝑋^∗=𝐴^𝑇𝐺𝐵^𝑇+𝐶^𝑇𝐺𝐷^𝑇+𝑃𝐴^𝑇𝐺𝐵^𝑇𝑄+𝑃𝐶^𝑇𝐺𝐷^𝑇𝑄. (2.24)
In the sequel, we will prove that 𝑋^∗ is just the least-norm solution of Problem 1. Consider the following minimum residual problemmin_{𝑋∈ℛ𝑟𝑚×𝑛()𝑃,𝑄}−𝐸𝐹𝐸𝐹.𝐴𝑋𝐵𝐶𝑋𝐷𝐴𝑃𝑋𝑄𝐵𝐶𝑃𝑋𝑄𝐷 (2.25)
Obviously, the solvability of Problem 1 is equivalent to that of the minimum residual problem (2.25), and the least-norm solution of Problem 1 must be the least-norm solution of the minimum residual problem (2.25).
In order to prove that 𝑋^∗ is the least-norm solution of Problem 1, it is enough to prove that 𝑋^∗ is the least-norm solution of the minimum residual problem (2.25). Denote vec(𝑋)=𝑥,vec(𝑋^∗)=𝑥^∗,vec(𝐺)=𝑔₁,vec(𝐺)=𝑔₂,vec(𝐸)=𝑒,vec(𝐹)=𝑓, then the minimum residual problem (2.25) is equivalent to the minimum residual problem as follows:min_{𝑥∈ℛ𝑚𝑛}𝐵^𝑇𝐷⊗𝐴_𝑇𝐵⊗𝐶_𝑇𝑄⊗()𝐷𝐴𝑃_𝑇𝑄⊗()𝑒𝑓𝑒𝑓.𝐶𝑃𝑥− (2.26) Noting that 𝑥^∗=vec𝐴^𝑇𝐺𝐵^𝑇+𝐶^𝑇𝐺𝐷^𝑇+𝑃𝐴^𝑇𝐺𝐵^𝑇𝑄+𝑃𝐶^𝑇𝐺𝐷^𝑇𝑄=𝐵⊗𝐴^𝑇𝑔₁+𝐷⊗𝐶^𝑇𝑔₂+(𝑄𝐵)⊗𝑃𝐴^𝑇𝑔₁+(𝑄𝐷)⊗𝑃𝐶^𝑇𝑔₂=𝐵⊗𝐴^𝑇𝐷⊗𝐶^𝑇(𝑄𝐵)⊗𝑃𝐴^𝑇(𝑄𝐷)⊗𝑃𝐶^𝑇𝑔₁𝑔₂𝑔₁𝑔₂=𝐵^𝑇𝐷⊗𝐴_𝑇𝐵⊗𝐶_𝑇𝑄⊗()𝐷𝐴𝑃_𝑇𝑄⊗(𝐶𝑃)^𝑇𝑔₁𝑔₂𝑔₁𝑔₂𝐵∈𝑅^𝑇𝐷⊗𝐴_𝑇𝐵⊗𝐶_𝑇𝑄⊗()𝐷𝐴𝑃_𝑇𝑄⊗(𝐶𝑃)^𝑇, (2.27) by Lemma 2.15 we can see that 𝑥^∗ is the least-norm solution of the minimum residual problem (2.26). Since vector operator is isomorphic, 𝑋^∗ is the unique least-norm solution of the minimum residual problem (2.25);furthermore 𝑋^∗ is the unique least-norm solution of Problem 1.

3. Solution of Problem 2

Since the solution set of Problem 1 is no empty, when 𝑋∈𝑆_𝐸, thenmin_{𝑋∈ℛ𝑟𝑚×𝑛()𝑃,𝑄}−𝐸𝐹𝐴𝑋𝐵𝐶𝑋𝐷⟺min_{𝑋∈ℛ𝑟𝑚×𝑛()𝑃,𝑄}𝐴𝑋−𝑋₀𝐵𝐶𝑋−𝑋₀𝐷−𝐸−𝐴𝑋₀𝐵𝐹−𝐶𝑋₀𝐷. (3.1)

Let 𝑋=𝑋−𝑋₀,𝐸=𝐸−𝐴𝑋₀𝐵,𝐹=𝐹−𝐶𝑋₀𝐷, then Problem 2 is equivalent to finding the least-norm generalized reflexive solution of a new corresponding minimum residual problem𝑋min_{∈ℛ𝑟𝑚×𝑛()𝑃,𝑄}𝐴𝐶−𝐸𝐹.𝑋𝐵𝑋𝐷 (3.2)

By using Algorithm 2.8, let initially iterative matrix 𝑋₁=𝐴^𝑇𝐻𝐵^𝑇+𝐶^𝑇𝐻𝐷^𝑇+𝑃𝐴^𝑇𝐻𝐵^𝑇𝑄+𝑃𝐶^𝑇𝐻𝐷^𝑇𝑄, or more especially, let 𝑋₁=0∈𝑅_𝑟^𝑚×𝑛(𝑃,𝑄); we can obtain the unique least-norm generalized reflexive solution 𝑋^∗ of minimum residual problem (3.2), then we can obtain the generalized reflexive solution 𝑋 of Problem 2, and 𝑋 can be represented that 𝑋=𝑋^∗+𝑋₀.

4. Numerical Examples

In this section, we will show several numerical examples to illustrate our results. All the tests are performed by MATLAB 7.8.

Example 4.1. Consider the generalized reflexive solution of Problem 1, where ,,.𝐴=13−57−92046−10−296−836227−13−55−22−1−1184−6−9−19,𝐵=408−54−150−234−10250392−6−27−8111𝐶=632−57−921046−119−1293−8136427−15−515−22−13−1129−6−9−19,𝐷=718−614−450−233−1208251694−6−58−2917𝐸=592−11911216−244−13313054311234−518221814−4071668−11765371434−1794083−1374−808242−3150−13621104−2848423−29091441−182−3326,𝐹=−288228302992291−48494096701090−783−7933363−1262979−385124626321734553−3709−100−1774−4534−45481256−6896864−2512−1136−1633−5412 (4.1) Let 𝑃=.000100000100−1001000001000,𝑄=0000−10001000−10001000−10000 (4.2)
We will find the least squares generalized reflexive solution of the matrix equation pair 𝐴𝑋𝐵=𝐸,𝐶𝑋𝐷=𝐹 by using Algorithm 2.8. Because of the influence of the error of calculation, ‖𝑃_𝑘‖=‖−∇𝐹(𝑋_𝑘)‖ is usually unequal to zero in the process of the iteration, where 𝑘=1,2,…. For any chosen positive number 𝜀, however small enough, for example, 𝜀=1.0000𝑒−8, whenever ‖𝑃_𝑘‖<𝜀, stop the iteration, and 𝑋_𝑘 is regarded to be the least squares generalized reflexive solution of the matrix equation pair 𝐴𝑋𝐵=𝐸,𝐶𝑋𝐷=𝐹.
Let𝑋₁=;0000000000000000000000000 (4.3) by Algorithm 2.8, we have the unique least Frobenius norm generalized reflexive solution of Problem 1𝑋₁₃₉=,𝑃5.00003.0000−6.000012.0000−5.0000−11.00008.0000−1.00009.00007.000013.0000−4.0000−8.00004.000013.00005.000012.00006.00003.0000−5.0000−7.00009.00001.00008.000011.0000₁₃₉=8.1258𝑒−009<𝜀,min_{𝑋∈ℛ𝑟𝑚×𝑛()𝑃,𝑄}−𝐸𝐹=𝐴𝑋𝐵𝐶𝑋𝐷𝐴𝑋₁₃₉𝐵𝐶𝑋₁₃₉𝐷−𝐸𝐹=1.9595𝑒−011. (4.4) The convergence curve for the Frobenius norm of the residual is shown in Figure 1.

Figure 1 

Convergence curve for the Frobenius norm of the residual for Example 4.1 with 𝑋1=𝟎.

Example 4.2. Consider the least-norm generalized reflexive solution of the minimum residual problem in Example 4.1. Let ,,𝑋𝐻=101020−10101−10012010−301210−10−2−10𝐻=−11−100010−131−10−202010−301210−10−212₁=𝐴^𝑇𝐻𝐵^𝑇+𝐶^𝑇𝐻𝐷^𝑇+𝑃𝐴^𝑇𝐻𝐵^𝑇𝑄+𝑃𝐶^𝑇𝐻𝐷^𝑇𝑄. (4.5) By using Algorithm 2.8, we have the unique least Frobenius norm generalized reflexive solution of Problem 1𝑋₁₁₈=,𝑃5.00003.0000−6.000012.0000−5.0000−11.00008.0000−1.00009.00007.000013.0000−4.0000−8.00004.000013.00005.000012.00006.00003.0000−5.0000−7.00009.00001.00008.000011.0000₁₁₈=9.8093𝑒−009<𝜀,min_{𝑋∈ℛ𝑟𝑚×𝑛()𝑃,𝑄}−𝐸𝐹=𝐴𝑋𝐵𝐶𝑋𝐷𝐴𝑋₁₁₈𝐵𝐶𝑋₁₁₈𝐷−𝐸𝐹=1.1235𝑒−010. (4.6)

The convergence curve for the Frobenius norm of the residual is shown in Figure 2.

Figure 2 

Convergence curve for the Frobenius norm of the residual for Example 4.2.

Example 4.3. Let 𝑆_𝐸 denote the set of all generalized reflexive solutions of Problem 1 in Example 4.1. For a given matrix 𝑋₀=−331110−7161010−90910−11−133−106−1−70∈ℛ_𝑟^5×5(𝑃,𝑄), (4.7) we will find 𝑋∈𝑆_𝐸, such that 𝑋−𝑋₀=min_{𝑋∈𝑆𝐸}𝑋−𝑋₀, (4.8) that is, find the optimal approximate solution to the matrix 𝑋₀ in 𝑆_𝐸.

Let 𝑋=𝑋−𝑋₀,𝐸=𝐸−𝐴𝑋₀𝐵,𝐹=𝐹−𝐶𝑋₀𝐷, by the method mentioned in Section 3, we can obtain the least-norm generalized reflexive solution 𝑋^∗ of the minimum residual problem (3.2) by choosing the initial iteration matrix 𝑋₁=𝟎, and 𝑋^∗ is such that𝑋^∗₁₀₂=,𝑃8.00000.0000−7.000011.0000−6.0000−11.000015.0000−2.00003.0000−3.00003.00005.0000−8.0000−5.00003.00006.000011.00007.00000.0000−8.00003.00003.00002.000015.000011.0000₁₀₂𝑋=8.6456𝑒−009<𝜀,𝑋=^∗₁₀₂+𝑋₀=.5.00003.0000−6.000012.0000−5.0000−11.00008.0000−1.00009.00007.000013.0000−4.0000−8.00004.000013.00005.000012.00006.00003.0000−5.0000−7.00009.00001.00008.000011.0000 (4.9)

The convergence curve for the Frobenius norm of the residual is shown in Figure 3.

Figure 3 

Convergence curve for the Frobenius norm of the residual for Example 4.3.

5. Conclusion

This paper mainly solves the minimum Frobenius norm residual problem and its optimal approximate problem over generalized reflexive matrices by constructing an iterative algorithm. We solve the minimum Frobenius norm residual problem by constructing an iterative algorithm, that is, for an arbitrary initial matrix 𝑋₁∈ℛ_𝑟^𝑚×𝑛(𝑃,𝑄),we obtain a solution 𝑋^∗∈ℛ_𝑟^𝑚×𝑛(𝑃,𝑄) of Problem 1 within finite iterative steps in the absence of round-off errors. The convergence of the algorithm is also proved. Let 𝑋₁=𝐴^𝑇𝐻𝐵^𝑇+𝐶^𝑇𝐻𝐷^𝑇+𝑃𝐴^𝑇𝐻𝐵^𝑇𝑄+𝑃𝐶^𝑇𝐻𝐷^𝑇𝑄, where 𝐻∈ℛ^𝑝×𝑞,𝐻∈ℛ^𝑠×𝑡 are arbitrary matrices, or more especially, let 𝑋₁=𝟎∈ℛ_𝑟^𝑚×𝑛(𝑃,𝑄); we obtain the unique least-norm solution 𝑋^∗ of the minimum Frobenius norm residual problem. Then we give the generalized reflexive solution of the optimal approximate problem by finding the least-norm generalized reflexive solution of a corresponding new minimum Frobenius norm residual problem.

Several numerical examples are given to confirm our theoretical results. We can see that our iterative algorithm is effective. We also note that for the minimum Frobenius norm residual problem with large but not sparse matrices 𝐴,𝐵,𝐶,𝐷,𝐸, and 𝐹, Algorithm 2.8 may be terminated more than 𝑚𝑛 steps because of computational errors.

Acknowledgments

This work was partially supported by the Research Fund Project (Natural Science 2010XJKYL018) of Sichuan University of Science and Engineering. This work is also supported by Open Fund of Geomathematics Key Laboratory of Sichuan Province (scsxdz2011005). Key Project of Natural Science of Educational Department in Sichuan Province (102A007).