An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations

Abstract

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations (𝐴𝑋𝐵−𝐶𝑌𝐷,𝐸𝑋𝐹−𝐺𝑌𝐻)=(𝑀,𝑁), which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matrices 𝑋 and 𝑌. When the matrix equations are consistent, for any initial generalized reflexive matrix pair [𝑋₁,𝑌₁], the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors, and the least Frobenius norm generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair [𝑋,𝑌] to a given matrix pair [𝑋₀,𝑌₀] in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair [𝑋^∗,𝑌^∗] of a new corresponding generalized coupled Sylvester matrix equation pair (𝐴𝑋𝐵−𝐶𝑌𝐷,𝐸𝑋𝐹−𝐺𝑌𝐻)=(𝑀,𝑁), where 𝑀=𝑀−𝐴𝑋₀𝐵+𝐶𝑌₀𝐷,𝑁=𝑁−𝐸𝑋₀𝐹+𝐺𝑌₀𝐻. Several numerical examples are given to show the effectiveness of the presented iterative algorithm.

1. Introduction

In this paper, the following notations are used. 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⟩=tr(𝐵^𝑇𝐴) for all 𝐴,𝐵∈ℛ^𝑚×𝑛, where tr(𝐴) denotes the trace of a matrix 𝐴. ‖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^𝑇𝑛)^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, diag(𝐴,𝐵) denotes the block diagonal matrix with 𝐴 and 𝐵 and being the main diagonal elements orderly. 𝐼_𝑛 denotes the n-order identity matrix.

Definition 1.1 (see [1, 2]). A matrix 𝑃∈ℛ^𝑛×𝑛 is said to be a generalized reflection matrix if 𝑃 satisfies that 𝑃^𝑇=𝑃,𝑃²=𝐼.

Definition 1.2 (see [1, 2]). Let 𝑃∈ℛ^𝑛×𝑛 and 𝑄∈ℛ^𝑛×𝑛 be two generalized reflection matrices. A matrix 𝐴∈ℛ^𝑛×𝑛 is called generalized reflexive (or generalized antireflexive) with respect to the matrix pair (𝑃,𝑄) if 𝑃𝐴𝑄=𝐴(or𝑃𝐴𝑄=−𝐴). The set of all n-by-n generalized reflexive matrices with respect to matrix pair (𝑃,𝑄) is denoted by ℛ^{𝑟𝑛×𝑛}(𝑃,𝑄).

The generalized reflexive and antireflexive matrices have many special properties and usefulness in engineering and scientific computations [1–6]. In particular, let 𝑃=𝑄, then a generalized reflexive matrix is called a reflexive matrix, which plays an important role in many areas and has been studied in [7–11]. Specially, let 𝑋^𝑇=𝑋, then a reflexive matrix 𝑋 is called a generalized bisymmetric matrix, which has been studied in [12, 13]. Moreover, let 𝑃=𝑄=𝐽_𝑛, then a generalized reflexive matrix is the well-known centrosymmetric matrix, which has been widely and extensively studied in [14–17].

The generalized coupled Sylvester systems play a fundamental role in the various fields of engineering theory, particularly in control systems. The numerical solution of the generalized coupled Sylvester systems has been addressed in a large body of literature. Kågström and Westin [18] developed a generalized Schur method by applying the QZ algorithm to solve (𝐴𝑋𝐵−𝐶𝑌𝐷,𝐸𝑋𝐹−𝐺𝑌𝐻)=(𝑀,𝑁). Ding and Chen [19] presented an iterative least squares solutions of (𝐴𝑋𝐵−𝐶𝑌𝐷,𝐸𝑋𝐹−𝐺𝑌𝐻)=(𝑀,𝑁) based on a hierarchical identification principle [20], in addition, by applying the hierarchical identification principle, Kılıçman and Zhour [21] developed an iterative algorithm for obtaining the weighted least-squares solution. Recently, some finite iterative algorithms have also been developed to solve matrix equations. For more detail, we refer to [11, 13, 22–30]. Wang [31, 32] gave the bi(skew)symmetric and centrosymmetric solutions to the system of quaternion matrix equations 𝐴₁𝑋=𝐶₁,𝐴₃𝑋𝐵₃=𝐶₃. Wang [33] also solved a system of matrix equations over arbitrary regular rings with identity. Chang and Wang [34] gave the necessary and sufficient conditions for the existence of and the expressions for the symmetric solutions of the matrix equations 𝐴𝑋+𝑌𝐴=𝐶,𝐴𝑋𝐴^𝑇+𝐵𝑌𝐵^𝑇=𝐶, and (𝐴^𝑇𝑋𝐴,𝐵^𝑇𝑋𝐵)=(𝐶,𝐷). Ding and Chen [25] also presented the gradient-based iterative algorithms by applying the gradient search principle and the hierarchical identification principle for the general coupled matrix equations ∑^𝑝𝑗=1𝐴_𝑖𝑗𝑋_𝑗𝐵_𝑖𝑗=𝑀_𝑖,𝑖=1,2,…,𝑝. Zhou et al. [35] proposed gradient-based iterative algorithms for solving the general coupled matrix equations with weighted least squares solutions. Wu et al. [36, 37] gave the finite iterative solutions to coupled Sylvester-conjugate matrix equations. Wu et al. [38] gave the finite iterative solutions to a class of complex matrix equations with conjugate and transpose of the unknowns. Jonsson and Kågström [39] proposed recursive block algorithms for solving the one-sided and coupled Sylvester matrix equations (𝐴𝑋−𝑌𝐵,𝐷𝑋−𝑌𝐸)=(𝐶,𝐹). Jonsson and Kågström [40] also proposed recursive block algorithms for the two-sided and generalized Sylvester and Lyapunov matrix equations. Dehghan and Hajarian [7, 8] gave the reflexive and generalized bisymmetric matrices solutions of the generalized coupled Sylvester matrix equations (𝐴𝑌−𝑍𝐵,𝐶𝑌−𝑍𝐷)=(𝐸,𝐹). Very recently, Dehghan and Hajarian [12] constructed an iterative algorithm to solve the generalized coupled Sylvester matrix equations (𝐴𝑋𝐵+𝐶𝑌𝐷,𝐸𝑋𝐹+𝐺𝑌𝐻)=(𝑀,𝑁) over generalized bisymmetric matrices. Huang et al. [13] present an iterative algorithm for the generalized coupled Sylvester matrix equations (𝐴𝑌−𝑍𝐵,𝐶𝑌−𝑍𝐷)=(𝐸,𝐹) and its optimal approximation problem over generalized reflexive matrices solutions. In [30], the similar but different iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations (𝐴𝑋𝐵−𝐶𝑌𝐷,𝐸𝑋𝐹−𝐺𝑌𝐻)=(𝑀,𝑁) and the optimal approximation problem over reflexive matrices. However, the generalized coupled Sylvester matrix equations (𝐴𝑋𝐵−𝐶𝑌𝐷,𝐸𝑋𝐹−𝐺𝑌𝐻)=(𝑀,𝑁) and the optimal approximation over generalized reflexive matrices have not been solved.

In this paper, we will consider the following two problems.

Problem 1. Let 𝑃∈ℛ^𝑚×𝑚,𝑄∈ℛ^𝑛×𝑛,𝑅∈ℛ^𝑠×𝑠, and 𝑆∈ℛ^𝑡×𝑡 be generalized reflection matrices. For given matrices 𝐴∈ℛ^𝑝×𝑚, 𝐵∈ℛ^𝑛×𝑞, 𝐶∈ℛ^𝑝×𝑠, 𝐷∈ℛ^𝑡×𝑞,𝑀∈ℛ^𝑝×𝑞, 𝐸∈ℛ^𝑘×𝑚, 𝐹∈ℛ^𝑛×𝑙, 𝐺∈ℛ^𝑘×𝑠, 𝐻∈ℛ^𝑡×𝑙, 𝑁∈ℛ^𝑘×𝑙, find a pair of matrices 𝑋∈ℛ^{𝑟𝑚×𝑛}(𝑃,𝑄), 𝑌∈ℛ^{𝑟𝑠×𝑡}(𝑅,𝑆) such that 𝐴𝐸𝑋𝑋𝐵𝐹−−𝐺𝐶𝑌𝑌𝐷𝐻==𝑀𝑁.,(1.1)

Problem 2. When Problem 1 is consistent, let 𝑆_𝐸 denote the set of the generalized reflexive solutions of Problem 1, that is, 𝑆_𝐸=[𝑋,𝑌]∣𝐴𝑋𝐵−𝐶𝑌𝐷=𝑀,𝐸𝑋𝐹−𝐺𝑌𝐻=𝑁,𝑌∈ℛ^{𝑟𝑚×𝑛}(𝑃,𝑄),𝑍∈ℛ^{𝑟𝑠×𝑡}(𝑅,𝑆).(1.2) For a given matrix pair [𝑌₀,𝑍₀]∈ℛ^{𝑟𝑚×𝑛}(𝑃,𝑄)×ℛ^{𝑟𝑠×𝑡}(𝑅,𝑆), find [𝑌,𝑍]∈𝑆_𝐸 such that ‖‖𝑌−𝑌₀‖‖²+‖‖𝑍−𝑍₀‖‖²=min_{[]𝑌,𝑍∈𝑆𝐸}‖‖𝑌−𝑌⁰‖‖²+‖‖𝑍−𝑍₀‖‖².(1.3)

The two-sided and generalized coupled Sylvester matrix equations (1.1) 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. In addition, as special type of generalized coupled Sylvester matrix equations (1.1), the generalized Sylvester matrix equation (𝐴𝑋−𝑌𝐵,𝐶𝑋−𝑌𝐷)=(𝐸,𝐹) arises in computing the deflating subspace of descriptor linear systems [18]. Wu et al. [36] presented some examples to show a motivation for studying (1.1). Problem 2 occurs frequently in experiment design, see for instance [41].

This paper is organized as follows. In Section 2, we will solve Problem 1 by constructing an iterative algorithm, that is, if Problem 1 is consistent, then for an arbitrary initial matrix pair [𝑌₁,𝑍₁]∈ℛ^{𝑟𝑚×𝑛}(𝑃,𝑄)×ℛ^{𝑟𝑠×𝑡}(𝑅,𝑆), we can obtain a solution pair [𝑌^∗,𝑍^∗] of Problem 1 within finite iterative steps in the absence of round-off errors. Let 𝑋₁=𝐴^𝑇𝐾𝐵^𝑇+𝐸^𝑇𝐿𝐹^𝑇+𝑃𝐴^𝑇𝐾𝐵^𝑇𝑄+𝑃𝐸^𝑇𝐿𝐹^𝑇𝑄 and 𝑌₁=−𝐶^𝑇𝐾𝐷^𝑇−𝐺^𝑇𝐿𝐻^𝑇−𝑅𝐶^𝑇𝐾𝐷^𝑇𝑆−𝑅𝐺^𝑇𝐿𝐻^𝑇𝑆, where 𝐾∈ℛ^𝑝×𝑞, 𝐿∈ℛ^𝑘×𝑙 are arbitrary matrices, or more especially, let 𝑋₁=0 and 𝑌₁=0, we can obtain the least Frobenius norm solutions of Problem 1. Then, in Section 3, we give the optimal approximate solution pair of Problem 2 by finding the least Frobenius norm generalized reflexive solution pair of the corresponding generalized coupled Sylvester matrix equations. In Section 4, several numerical examples are given to illustrate the application of our method. At last, some conclusions are drawn in Section 5.

2. An Iterative Algorithm for Solving Problem 1

In this section, we will first introduce an iterative algorithm to solve Problem 1, then prove that it is convergent. Then, we will give the least-norm generalized reflexive solutions of Problem 1 when an appropriate initial iterative matrix pair is chosen.

For the purpose of simplification, we introduce the following operators:ΦΨ((𝑋𝑋,,𝑌𝑌))==𝐴𝐸𝑋𝑋𝐹𝐵−−𝐺𝐶𝑌𝑌𝐻𝐷,.(2.1)

Algorithm 2.1. We have the following steps.

Step 1. Input matrices 𝐴∈ℛ^𝑝×𝑚, 𝐵∈ℛ^𝑛×𝑞, 𝐶∈ℛ^𝑝×𝑠, 𝐷∈ℛ^𝑡×𝑞, 𝑀∈ℛ^𝑝×𝑞, 𝐸∈ℛ^𝑘×𝑚, 𝐹∈ℛ^𝑛×𝑙, 𝐺∈ℛ^𝑘×𝑠, 𝐻∈ℛ^𝑡×𝑙, 𝑁∈ℛ^𝑘×𝑙, and four generalized reflection matrices 𝑃∈ℛ^𝑚×𝑚, 𝑄∈ℛ^𝑛×𝑛, 𝑅∈ℛ^𝑠×𝑠,𝑆∈ℛ^𝑡×𝑡.

Step 2. Choose two arbitrary matrices 𝑋₁∈ℛ^{𝑟𝑚×𝑛}(𝑃,𝑄), 𝑌₁∈ℛ^{𝑟𝑠×𝑡}(𝑅,𝑆). Compute 𝑅𝑈₁=diag𝑀−Φ𝑋₁,𝑌₁,𝑁−Ψ𝑋₁,𝑌₁₁,=12𝐴^𝑇𝑀−Φ𝑋₁,𝑌₁𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐹^𝑇𝑉₁+𝑃𝐴^𝑇𝑀−Φ𝑋₁,𝑌₁𝐵^𝑇𝑄+𝑃𝐸^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐹^𝑇𝑄,=12−𝐶^𝑇𝑀−Φ𝑋₁,𝑌₁𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐻^𝑇𝑘∶=1−.𝑅𝐶^𝑇𝑀−Φ𝑋₁,𝑌₁𝐷^𝑇𝑆−𝑅𝐺^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐻^𝑇𝑆,(2.2)

Step 3. If 𝑅_𝑘=0, then stop and [𝑋_𝑘,𝑌_𝑘] is the solution of the generalized coupled Sylvester matrix equation (1.1); else if 𝑅_𝑘≠0, but 𝑈_𝑘=0 and 𝑉_𝑘=0, then stop and the generalized coupled Sylvester matrix equations (1.1) are not consistent over generalized reflexive matrices; else 𝑘∶=𝑘+1.

Step 4. Compute 𝑋_𝑘=𝑋_𝑘−1+‖‖𝑅^𝑘−1‖‖²‖‖𝑈_𝑘−1‖‖²+‖‖𝑉_𝑘−1‖‖²𝑌_𝑘𝑈_𝑘−1,=𝑌_𝑘−1+‖‖𝑅^𝑘−1‖‖²‖‖𝑈_𝑘−1‖‖²+‖‖𝑉_𝑘−1‖‖²𝑅_𝑘𝑉_𝑘−1,==d𝑅iag_𝑘−1𝑀−Φ𝑋_𝑘,𝑌_𝑘,𝑁−Ψ𝑋_𝑘,𝑌_𝑘−‖‖𝑅^𝑘−1‖‖²‖‖𝑈_𝑘−1‖‖²+‖‖𝑉_𝑘−1‖‖²𝑈diagΦ𝑈_𝑘−1,𝑉_𝑘−1,Ψ𝑈_𝑘−1,𝑉_𝑘−1_𝑘,=12𝐴^𝑇𝑀−Φ𝑋_𝑘,𝑌_𝑘𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋_𝑘,𝑌_𝑘𝐹^𝑇+𝑃𝐴^𝑇𝑀−Φ𝑋_𝑘,𝑌_𝑘𝐵^𝑇𝑄+𝑃𝐸^𝑇𝑁−Ψ𝑋_𝑘,𝑌_𝑘𝐹^𝑇𝑄+‖‖𝑅^𝑘‖‖²‖‖𝑅_𝑘−1‖‖²𝑉_𝑘𝑈_𝑘−1,=12−𝐶^𝑇𝑀−Φ𝑋_𝑘−1,𝑌_𝑘−1𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋_𝑘−1,𝑌_𝑘−1𝐻^𝑇−𝑅𝐶^𝑇𝑀−Φ𝑋_𝑘−1,𝑌_𝑘−1𝐷^𝑇𝑆−𝑅𝐺^𝑇𝑁−Ψ𝑋_𝑘−1,𝑌_𝑘−1𝐻^𝑇𝑆+‖‖𝑅^𝑘‖‖²‖‖𝑅_𝑘−1‖‖²𝑉_𝐾−1.(2.3)

Step 5. Go to Step 3.
Obviously, it can be seen that 𝑋_𝑘,𝑈_𝑘∈𝑅^{𝑟𝑚×𝑛}(𝑃,𝑄), 𝑌_𝑘,𝑉_𝑘∈𝑅^{𝑟𝑠×𝑡}(𝑅,𝑆), where 𝑘=1,2,….

Lemma 2.2. For the sequences {𝑅_𝑖}, {𝑈_𝑖}, and {𝑉_𝑖} generated by Algorithm 2.1, and 𝑠≥2, we havetr𝑅^𝑇𝑖𝑅_𝑗=0,tr𝑈^𝑇𝑖𝑈_𝑗+𝑉^𝑇𝑖𝑉_𝑗=0,𝑖,𝑗=1,2,…,𝑠,𝑖≠𝑗.(2.4)

The proof of Lemma 2.2 is presented in Appendix A.

Lemma 2.3. Suppose [𝑋^∗,𝑌^∗] is an arbitrary solution pair of Problem 1, then for any initial generalized reflexive matrix pair [𝑋₁,𝑌₁], we have tr𝑋^∗−𝑋_𝑖^𝑇𝑈_𝑖+𝑌^∗−𝑌_𝑖^𝑇𝑉_𝑖=‖‖𝑅^𝑖‖‖²,𝑘=1,2,…,(2.5) where the sequences {𝑋_𝑖},{𝑌_𝑖}, {𝑈_𝑖}, {𝑉_𝑖}, and {𝑅_𝑖} are generated by Algorithm 2.1.

The proof of Lemma 2.3 is presented in Appendix B.

Remark 2.4. If there exist, a positive number 𝑘 such that 𝑈_𝑘=0 and 𝑉_𝑘=0 but 𝑅_𝑘≠0, then by Lemma 2.3, we have that the generalized coupled Sylvester matrix equations (1.1) are not consistent over generalized reflexive matrices.

Theorem 2.5. Suppose that Problem 1 is consistent, then for an arbitrary initial matrix pair [𝑋₁,𝑌₁]∈ℛ^{𝑟𝑚×𝑛}(𝑃,𝑄)×ℛ^{𝑟𝑠×𝑡}(𝑅,𝑆), a generalized reflexive solution pair of Problem 1 can be obtained with finite iteration steps in the absence of round-off errors.

Proof. If 𝑅_𝑖≠𝟎,𝑖=1,2,…,𝑝𝑞+𝑠𝑡, by Lemma 2.3, we have 𝑈_𝑖≠𝟎,𝑉_𝑖≠𝟎,𝑖=1,2,…,𝑝𝑞+𝑠𝑡, then we can compute [𝑋_{𝑝𝑞+𝑠𝑡+1},𝑌_{𝑝𝑞+𝑠𝑡+1}] by Algorithm 2.1.
By Lemma 2.2, we havetrtr𝑅𝑅^{𝑇𝑝𝑞+𝑠𝑡+1}_𝑇𝑖𝑅_𝑖=0,𝑖=1,2,…,𝑝𝑞+𝑠𝑡,𝑅_𝑗=0,𝑖,𝑗=1,2,…,𝑝𝑞+𝑠𝑡,𝑖≠𝑗.(2.6)
It can be seen that the set of 𝑅₁,𝑅₂,…,𝑅_{𝑝𝑞+𝑠𝑡} is an orthogonal basis of the matrix subspace𝑆=𝐿∣𝐿=diag𝐿₁,𝐿₂,𝐿₁∈ℛ^𝑝×𝑞,𝐿₂∈ℛ^𝑠×𝑡,(2.7) which implies that 𝑅_{𝑝𝑞+𝑠𝑡+1}=0, that is, [𝑋_{𝑝𝑞+𝑠𝑡+1},𝑌_{𝑝𝑞+𝑠𝑡+1}]∈ℛ^{𝑟𝑚×𝑛}(𝑃,𝑄)×ℛ^{𝑟𝑠×𝑡}(𝑅,𝑆) is a solution pair of Problem 1. This completes the proof.

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

Lemma 2.6 (see [42, Lemma 2.4]). Suppose that the consistent system of linear equation 𝐴𝑥=𝑏 has a solution 𝑥^∗∈𝑅(𝐴^𝑇), then 𝑥^∗ is a unique least Frobenius norm solution of the system of linear equation.

By Lemma 2.6, the following result can be obtained.

Theorem 2.7. Suppose that Problem 1 is consistent. If we choose the initial iterative matrices 𝑋₁=𝐴^𝑇𝐾𝐵^𝑇+𝐸^𝑇𝐿𝐹^𝑇+𝑃𝐴^𝑇𝐾𝐵^𝑇𝑄+𝑃𝐸^𝑇𝐿𝐹^𝑇𝑄 and 𝑌₁=−𝐶^𝑇𝐾𝐷^𝑇−𝐺^𝑇𝐿𝐻^𝑇−𝑅𝐶^𝑇𝐾𝐷^𝑇𝑆 −𝑅𝐺^𝑇𝐿𝐻^𝑇𝑆, where 𝐾∈ℛ^𝑝×𝑞,𝐿∈ℛ^𝑘×𝑙 are arbitrary matrices, especially, 𝑋₁=0∈ℛ^𝑚×𝑛(𝑃,𝑄) and 𝑌₁=0∈ℛ^𝑠×𝑡(𝑅,𝑆), then the solution pair [𝑌^∗,𝑍^∗] generated by Algorithm 2.1 is the unique least Frobenius norm generalized reflexive solutions of Problem 1.

Proof. We know the solvability of the generalized coupled Sylvester matrix equations (1.1) over generalized reflexive matrices is equivalent to the following matrix equations: 𝐴𝐸𝑃𝑃𝑋𝑋𝐴𝐸𝑋𝑄𝑋𝑄𝐵𝐵𝐹𝐹−−−−𝐺𝐺𝐶𝐶𝑅𝑌𝑅𝑌𝐷𝐻𝑌𝑌𝑆𝑆==𝐷𝐻𝑀𝑁==,,𝑀𝑁.,(2.8) Then, the system of matrix equations (2.8) is equivalent to ⎛⎜⎜⎜⎜⎜⎜⎝𝐹𝐵^𝑇⊗𝐴−𝐷^𝑇⊗𝐶_𝑇𝐵⊗𝐸−𝐻^𝑇⊗𝐺_𝑇𝐹𝑄⊗𝐴𝑃−𝐷^𝑇𝑆⊗𝐶𝑅_𝑇𝑄⊗𝐸𝑃−𝐻^𝑇𝑆⊗𝐺𝑅⎞⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎝vveecc((𝑋𝑌))⎞⎟⎟⎠=⎛⎜⎜⎜⎜⎜⎜⎝vvvveeeecccc((((𝑀𝑀𝑁𝑁))))⎞⎟⎟⎟⎟⎟⎟⎠.(2.9) Let 𝑋₁=𝐴^𝑇𝐾𝐵^𝑇+𝐸^𝑇𝐿𝐹^𝑇+𝑃𝐴^𝑇𝐾𝐵^𝑇𝑄+𝑃𝐸^𝑇𝐿𝐹^𝑇𝑄 and 𝑌₁=−𝐶^𝑇𝐾𝐷^𝑇−𝐺^𝑇𝐿𝐻^𝑇 − 𝑅𝐶^𝑇𝐾𝐷^𝑇𝑆−𝑅𝐺^𝑇𝐿𝐻^𝑇𝑆, where 𝐾∈ℛ^𝑝×𝑞,𝐿∈ℛ^𝑘×𝑙 are arbitrary matrices, then ⎛⎜⎜⎝vveecc𝑋𝑌₁₁⎞⎟⎟⎠=⎛⎜⎜⎝vec𝐴^𝑇vec−𝐶_𝑇𝐾𝐵^𝑇+𝐸^𝑇𝐿𝐹^𝑇+𝑃𝐴^𝑇𝐾𝐵^𝑇𝑄+𝑃𝐸^𝑇𝐿𝐹^𝑇𝑄=⎛⎜⎜⎝𝐵⊗𝐴_𝑇𝐾𝐷^𝑇−𝐺^𝑇𝐿𝐻^𝑇−𝑅𝐶^𝑇𝐾𝐷^𝑇𝑆−𝑅𝐺^𝑇𝐿𝐻^𝑇𝑆⎞⎟⎟⎠𝐹⊗𝐸^𝑇𝑄𝐵⊗𝑃𝐴^𝑇𝑄𝐹⊗𝑃𝐸^𝑇−𝐷⊗𝐶^𝑇−𝐻⊗𝐺^𝑇−𝑆𝐷⊗𝑅𝐶^𝑇−𝑆𝐻⊗𝑅𝐺^𝑇=⎛⎜⎜⎜⎜⎜⎜⎝𝐵⎞⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎝vvvveeeecccc((((𝐾𝐾𝐿𝐿))_𝑇))⎞⎟⎟⎟⎟⎟⎟⎠𝐹⊗𝐴−𝐷^𝑇⊗𝐶_𝑇𝐵⊗𝐸−𝐻^𝑇⊗𝐺_𝑇𝐹𝑄⊗𝐴𝑃−𝐷^𝑇𝑆⊗𝐶𝑅_𝑇𝑄⊗𝐸𝑃−𝐻^𝑇𝑆⊗𝐺𝑅⎞⎟⎟⎟⎟⎟⎟⎠^𝑇∈ℛ⎛⎜⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎜⎜⎝𝐵_𝑇⎛⎜⎜⎜⎜⎜⎜⎝vvvveeeecccc((((𝐾𝐾𝐺𝐺))))⎞⎟⎟⎟⎟⎟⎟⎠𝐹⊗𝐴−𝐷^𝑇⊗𝐶_𝑇𝐵⊗𝐸−𝐻^𝑇⊗𝐺_𝑇𝐹𝑄⊗𝐴𝑃−𝐷^𝑇𝑆⊗𝐶𝑅_𝑇𝑄⊗𝐸𝑃−𝐻^𝑇𝑆⊗𝐺𝑅⎞⎟⎟⎟⎟⎟⎟⎠^𝑇⎞⎟⎟⎟⎟⎟⎟⎟⎠.(2.10)Furthermore, we can see that all 𝑋_𝑘,𝑌_𝑘 generated by Algorithm 2.1 satisfy ⎛⎜⎜⎝vveecc𝑋𝑌_𝑘𝑘⎞⎟⎟⎠∈ℛ⎛⎜⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎜⎜⎝𝐵^𝑇𝐹⊗𝐴−𝐷^𝑇⊗𝐶_𝑇𝐵⊗𝐸−𝐻^𝑇⊗𝐺_𝑇𝐹𝑄⊗𝐴𝑃−𝐷^𝑇𝑆⊗𝐶𝑅_𝑇𝑄⊗𝐸𝑃−𝐻^𝑇𝑆⊗𝐺𝑅⎞⎟⎟⎟⎟⎟⎟⎠^𝑇⎞⎟⎟⎟⎟⎟⎟⎟⎠,(2.11)by Lemma 2.6, we know that [𝑋^∗,𝑌^∗] is the least Frobenius norm generalized reflexive solution pair of the system of linear equations (2.9). Since vector operator is isomorphic, [𝑋^∗,𝑌^∗] is the unique least Frobenius norm generalized reflexive solution pair of the system of matrix equations (2.8), then [𝑋^∗,𝑌^∗] is the unique least Frobenius norm generalized reflexive solution pair of Problem 1.

3. The Solution of Problem 2

In this section, we will show that the optimal approximate solutions of Problem 2 for a given generalized reflexive matrix pair can be derived by finding the least Frobenius norm generalized reflexive solutions of the corresponding generalized coupled Sylvester matrix equations.

When Problem 1 is consistent, the set of generalized reflexive solutions of Problem 1 denoted by 𝑆_𝐸 is not empty. For a given matrix pair [𝑋₀,𝑌₀]∈ℛ^{𝑟𝑚×𝑛}(𝑃,𝑄)×ℛ^{𝑟𝑠×𝑡}(𝑅,𝑆), we have𝐴𝐸𝑋𝑋𝐹𝐵−−𝐺𝐶𝑌𝑌𝐻𝐷==𝑀𝑁⟺𝐴𝑋−𝑋⁰𝐸𝑋−𝑋₀𝐵−𝐶𝑌−𝑌₀𝐷=𝑀−𝐴𝑋₀𝐵+𝐶𝑌₀𝐷𝐹−𝐺𝑌−𝑌₀𝐻=𝑁−𝐸𝑋₀𝐹+𝐺𝑌₀𝐻(3.1) Set 𝑋=𝑋−𝑋₀,𝑌=𝑌−𝑌₀,𝑀=𝑀−𝐴𝑋₀𝐵+𝐶𝑌₀𝐷,𝑁=𝑁−𝐸𝑋₀𝐹+𝐺𝑌₀𝐻, then Problem 2 is equivalent to that of finding the least Frobenius norm generalized reflexive solutions pair [𝑋^∗,𝑌^∗] of the corresponding generalized coupled Sylvester matrix equations𝐴𝐸𝑋𝑋𝐵𝐹−−𝐶𝐺𝑌𝑌𝐷𝐻==𝑀𝑁.,(3.2) By using Algorithm 2.1, let initial iteration matrix 𝑋₁=𝐴^𝑇𝐾𝐵^𝑇+𝐸^𝑇𝐿𝐹^𝑇+𝑃𝐴^𝑇𝐾𝐵^𝑇𝑄+𝑃𝐸^𝑇𝐿𝐹^𝑇𝑄 and 𝑌₁=−𝐶^𝑇𝐾𝐷^𝑇−𝐺^𝑇𝐿𝐻^𝑇−𝑅𝐶^𝑇𝐾𝐷^𝑇𝑆−𝑅𝐺^𝑇𝐿𝐻^𝑇𝑆, or more especially, let 𝑋₁=0∈ℛ^{𝑟𝑚×𝑛}(𝑃,𝑄) and 𝑌₁=0∈ℛ^{𝑟𝑠×𝑡}(𝑅,𝑆), then we can get the least Frobenius norm generalized reflexive solution pair [𝑋^∗,𝑌^∗] of (3.2). Thus, the generalized reflexive solution pair of the problem 2 can be represented as [𝑋,𝑌]=[𝑋^∗+𝑋₀,𝑌^∗+𝑌₀].

4. Numerical Experiments

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 solutions of the generalized coupled Sylvester matrix equations 𝐴𝑋𝐵−𝐶𝑌𝐷=𝑀,𝐸𝑋𝑌−𝐺𝑌𝐻=𝑁, where 𝑀=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−−−−518157112015963434𝐴𝐶𝐸𝐺1111==288511==530837⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝890327−⎛⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎜⎜⎝−1120386292−−−−1135512−−211432853867014−−5−530648021543109572484962131950352−−−−−−−2−−−221391251−4924257−6334567207634213085−668375428282−13−−38⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠766216−−−19−−−,4816⎞⎟⎟⎟⎟⎟⎟⎠111⎞⎟⎟⎟⎟⎟⎟⎠989,−511−−−−−,⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠1918391,𝑁⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝐻=𝐹𝐷⎛⎜⎜⎜⎜⎜⎜⎝==−−−=−⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝211⎛⎜⎜⎜⎜⎜⎜⎝𝐵276516436−⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝−15238=1−−1−−3961473143414512⎛⎜⎜⎜⎜⎜⎜⎝−−−135262943−−561246859758161453−−−7472795789565−−−−2−−−−−−−5282−809−12−822222285299−61714316−−57365712398493143066⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎟⎟⎠,,21,29,0799951226−767780168325⎞⎟⎟⎟⎟⎟⎟⎠.(4.1)Let 𝑃=𝑅⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝=−00001⎛⎜⎜⎜⎜⎜⎜⎝0001000100100−00001−0001010001000−⎞⎟⎟⎟⎟⎟⎟⎠00001,⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑆=𝑄⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝=00010⎛⎜⎜⎜⎜⎜⎜⎝000010010−−00000001110000100001000−000⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠1⎞⎟⎟⎟⎟⎟⎟⎠,(4.2)be generalized reflection matrices.
We will find the generalized reflexive solutions of the matrix equations 𝐴𝑋𝐵−𝐶𝑌𝐷=𝑀,𝐸𝑋𝑌−𝐺𝑌𝐻=𝑁 by using Algorithm 2.1. It can be verified that the matrix equations are consistent over generalized reflexive matrices and the solutions are𝑋^∗=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝−−13712219513−132217−−59311⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑌^∗=⎛⎜⎜⎜⎜⎜⎜⎝−19343−17468−−01181339417486⎞⎟⎟⎟⎟⎟⎟⎠.(4.3) Because of the influence of the error of calculation, the residual 𝑅_𝑖 is usually unequal to zero in the process of the iteration, where 𝑖=1,2,…. For any chosen positive number 𝜀; however, small enough, for example, 𝜀=1.0000𝑒−010, whenever ‖𝑅_𝑘‖<𝜀, stop the iteration, 𝑋_𝑘 and 𝑌_𝑘 are regarded to be generalized reflexive solutions of the matrix equations 𝐴𝑋𝐵−𝐶𝑌𝐷=𝑀,𝐸𝑋𝑌−𝐺𝑌𝐻=𝑁. Choose an initially iterative matrix pair [𝑋₁,𝑌₁]∈ℛ^𝑟5×4(𝑃,𝑄)×ℛ^𝑟4×5(𝑅,𝑆), such as 𝑋₁=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝−−67312−−24812−2136742281⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑌₁=⎛⎜⎜⎜⎜⎜⎜⎝−39334171−−01183933−7141⎞⎟⎟⎟⎟⎟⎟⎠.(4.4) By Algorithm 2.1, we have 𝑌𝑋₃₀=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝−−137122..00...00000000000000000013195.....00000000000000000000−122317...00.0.00000000000000000−−59311...000..00000000000000₃₀000⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,=⎛⎜⎜⎜⎜⎜⎜⎝−19343...00.0000000000‖‖0000𝑅−17468..00..00000000000000−−118.0..00000000000013394...000.000000000000018746...0.000000000000000₃₀⎞⎟⎟⎟⎟⎟⎟⎠,‖‖=2.9703𝑒−012<𝜀.(4.5)So we obtain the generalized reflexive solutions of the matrix equations 𝐴𝑋𝐵−𝐶𝑌𝐷=𝑀,𝐸𝑋𝑌−𝐺𝑌𝐻=𝑁. The relative error of the solutions and the residual are shown in Figure 1, where the relative error 𝑅𝐸𝑘=(‖𝑋_𝑘−𝑋^∗‖+‖𝑌_𝑘−𝑌^∗‖)/(‖𝑋^∗‖+‖𝑌^∗‖) and the residual 𝑅𝑘=‖𝑅_𝑘‖.
Let𝑋₁=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝00000000000000000000⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑌₁=⎛⎜⎜⎜⎜⎜⎜⎝00000000000000000000⎞⎟⎟⎟⎟⎟⎟⎠,(4.6) by Algorithm 2.1, we have 𝑌𝑋^∗=𝑋₃₀=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝−−137122..00...00000000000000000095131.....00000000000000000000−132217...000..00000000000000000−−59311...000..00000000000000_∗000⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,=𝑌₃₀=⎛⎜⎜⎜⎜⎜⎜⎝−19343...00.‖‖0000𝑅0000000000−17468..00..00000000000000−−118.0..00000000000013394...000.000000000000014876...000.0000000000000₃₀⎞⎟⎟⎟⎟⎟⎟⎠,‖‖=8.2565𝑒−012<𝜀.(4.7)The relative error of the solutions and the residual are shown in Figure 2.

Figure 1 

The relative error of the solutions and the residual for Example 4.1 with 𝑋1≠0,𝑌1≠0.

Figure 2 

The relative error of the solutions and the residual for Example 4.1 with 𝑋1=0,𝑌1=0.

Example 4.2. Consider the unique least-norm generalized reflexive solutions of the matrix equations in Example 4.1. Let 𝑋𝐾=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−101201−−000111−100122−−211113⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝐿=⎛⎜⎜⎜⎜⎜⎜⎝−0121−1101−−−1112−03035236₁⎞⎟⎟⎟⎟⎟⎟⎠,𝑌=𝐴^𝑇𝐾𝐵^𝑇+𝐶^𝑇𝐿𝐷^𝑇+𝑃𝐴^𝑇𝐾𝐵^𝑇𝑄+𝑃𝐶^𝑇𝐿𝐷^𝑇𝑄,₁=−𝐸^𝑇𝐾𝐹^𝑇−𝐺^𝑇𝐿𝐻^𝑇−𝑅𝐸^𝑇𝐾𝐹^𝑇𝑆−𝑅𝐺^𝑇𝐿𝐻^𝑇𝑆.(4.8)By using Algorithm 2.1, we have the least-norm generalized reflexive solutions of the matrix equations 𝐴𝑋𝐵−𝐶𝑌𝐷=𝑀,𝐸𝑋𝑌−𝐺𝑌𝐻=𝑁 as follows: 𝑌𝑋^∗=𝑋₃₀=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝−−137122...00..00000000000000000019513.....00000000000000000000−132217...000..00000000000000000−−59311...000..00000000000000_∗000⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,=𝑌₃₀=⎛⎜⎜⎜⎜⎜⎜⎝−19343..00..00000000000000−17468..‖‖.00.0𝑅0000000000000−−118.0..00000000000013394...000.000000000000014876...000.0000000000000₃₀⎞⎟⎟⎟⎟⎟⎟⎠,‖‖=2.3986𝑒−011𝑒−012<𝜀.(4.9)
The relative error of the solutions and the residual are shown in Figure 3.

Figure 3 

The relative error of the solutions and the residual for Example 4.2.

Example 4.3. Let 𝑆_𝐸 denote the set of all generalized reflexive solutions of the matrix equations in Example 4.1. For a given generalized reflexive matrices 𝑋₀=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝−33102−−−20123−−20313−−20231⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑌₀=⎛⎜⎜⎜⎜⎜⎜⎝2152−4302−0222−21250324⎞⎟⎟⎟⎟⎟⎟⎠,(4.10)we will find [𝑋,𝑌]∈𝑆_𝐸, such that ‖‖𝑋−𝑋₀‖‖+‖‖𝑌−𝑌₀‖‖=min_{[]𝑋,𝑌∈𝑆𝐸}‖‖𝑋−𝑋₀‖‖+‖‖𝑌−𝑌₀‖‖,(4.11)that is, find the optimal approximate generalized reflexive solution pair to the matrix pair [𝑋₀,𝑌₀] in 𝑆_𝐸 in Frobenius norm.
Let 𝑋=𝑋−𝑋₀,𝑌=𝑌−𝑌₀,𝑀=𝑀−𝐴𝑋₀𝐵+𝐶𝑌₀𝐷,𝑁=𝑁−𝐸𝑋₀𝐹+𝐺𝑌₀𝐻, by the method mentioned in Section 3, we can obtain the least-norm generalized reflexive solution pair [𝑋^∗,𝑌^∗] of the matrix equations 𝐴𝑋𝐵+𝐶𝑌𝐷=𝑀,𝐸𝑋𝐹+𝐺𝑌𝐻=𝑁 by choosing the initial iteration matrices 𝑋₁=0 and 𝑌₁=0, then by Algorithm 2.1, we have that𝑌𝑋^∗=𝑋^∗30=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝−−−161500..0...0000000000000000000113360....0.0000000000000000000−−105160..0.0..000000000000000000−−136031..0.0..0000000000000000_∗00⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,=𝑌^∗30‖‖𝑅=⎛⎜⎜⎜⎜⎜⎜⎝−18128...00.00000000000000−14426...00.00000000000000−−111.00.0.0000000000018182...0.00000000000000014462...0.000000000000000₃₀⎞⎟⎟⎟⎟⎟⎟⎠,‖‖=6.3482𝑒−010<𝜀=1.0000𝑒−010(4.12)and the optimal approximate generalized reflexive solutions to the matrix pair [𝑋₀,𝑌₀] in Frobenius norm are 𝑌=𝑋𝑌=_∗30𝑋^∗30+𝑋₀=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝−−137122..00...00000000000000000013195.....00000000000000000000−122317...00.0.00000000000000000−−59311...000..00000000000000000⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,+𝑌₀=⎛⎜⎜⎜⎜⎜⎜⎝−19343...00.00000000000000−17468...00.00000000000000−−118.0..00000000000019334...0.00000000000000017486...0.000000000000000⎞⎟⎟⎟⎟⎟⎟⎠.(4.13)The relative error of the solutions and the residual are shown in Figure 4, where the relative error 𝑅𝐸𝑘=(‖𝑋_𝑘+𝑋₀−𝑋^∗‖+‖𝑌_𝑘+𝑌₀−𝑌^∗‖)/(‖𝑋^∗‖+‖𝑌^∗‖) and the residual 𝑅𝑘=‖𝑅_𝑘‖.

Figure 4 

The relative error of the solutions and the residual for Example 4.3.

5. Conclusions

In this paper, an efficient iterative algorithm is presented to solve the generalized coupled Sylvester matrix equations 𝐴𝑋𝐵−𝐶𝑌𝐷=𝑀,𝐸𝑋𝑌−𝐺𝑌𝐻=𝑁 over generalized reflexive matrix pair [𝑋,𝑌]∈ℛ^{𝑟𝑚×𝑛}(𝑃,𝑄)×ℛ^{𝑟𝑠×𝑡}(𝑅,𝑆). When the matrix equations 𝐴𝑋𝐵−𝐶𝑌𝐷=𝑀,𝐸𝑋𝑌−𝐺𝑌𝐻=𝑁 are consistent over generalized reflexive matrices 𝑋 and 𝑌, for any generalized reflexive initial iterative matrix pair [𝑋₁,𝑌₁]∈ℛ^{𝑟𝑚×𝑛}(𝑃,𝑄)×ℛ^{𝑟𝑠×𝑡}(𝑅,𝑆), the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors. Let initial matrices 𝑋₁=𝐴^𝑇𝐾𝐵^𝑇+𝐸^𝑇𝐿𝐹^𝑇+𝑃𝐴^𝑇𝐾𝐵^𝑇𝑄+𝑃𝐸^𝑇𝐿𝐹^𝑇𝑄 and 𝑌₁=−𝐶^𝑇𝐾𝐷^𝑇−𝐺^𝑇𝐿𝐻^𝑇−𝑅𝐶^𝑇𝐾𝐷^𝑇𝑆−𝑅𝐺^𝑇𝐿𝐻^𝑇𝑆, where 𝐾∈ℛ^𝑝×𝑞,𝐿∈ℛ^𝑘×𝑙 are arbitrary matrices, especially, let 𝑋₁=0∈ℛ^{𝑟𝑚×𝑛}(𝑃,𝑄) and 𝑌₁=0∈ℛ^{𝑟𝑠×𝑡}(𝑅,𝑆), the unique least-norm generalized reflexive solutions of the matrix equations can be derived. Furthermore, the optimal approximate solutions of 𝐴𝑋𝐵−𝐶𝑌𝐷=𝑀,𝐸𝑋𝑌−𝐺𝑌𝐻=𝑁 for a given generalized reflexive matrix pair [𝑋₀,𝑌₀]∈ℛ^{𝑟𝑚×𝑛}(𝑃,𝑄)×ℛ^{𝑟𝑠×𝑡}(𝑅,𝑆) can be derived by finding the least-norm generalized reflexive solutions of two new corresponding generalized coupled Sylvester matrix equations. Finally, several numerical examples are given to illustrate that our iterative algorithm is quite effective.

The results presented in this paper generalize some previous results [7, 12, 13, 30]. When 𝐵=𝐼,𝐶=𝐼,𝐹=𝐼,𝐺=𝐼, 𝑃=𝑄, and 𝑅=𝑆, then our results reduce to those in [7]. When 𝑃=𝑄,𝑅=𝑆,𝑋^𝑇=𝑋, and 𝑌^𝑇=𝑌, the results in this paper reduce to those in [12]. When 𝐵=𝐼,𝐶=𝐼,𝐹=𝐼, and 𝐺=𝐼, then the results in this paper reduce to those in [13]. When 𝑃=𝑄 and 𝑅=𝑆, then the results in this paper reduce to those in [30].

Appendices

A. The Proof of Lemma 2.2

Since tr(𝑅^𝑇𝑖𝑅_𝑗)=tr(𝑅^𝑇𝑗𝑅_𝑖), tr(𝑈^𝑇𝑖𝑈_𝑗)=tr(𝑈^𝑇𝑗𝑈_𝑖), and tr(𝑉^𝑇𝑖𝑉_𝑗)=tr(𝑉^𝑇𝑗𝑉_𝑖) for all 𝑖,𝑗=1,2,…,𝑠, we only need to prove thattr𝑅^𝑇𝑖𝑅_𝑗=0,tr𝑈^𝑇𝑖𝑈_𝑗+𝑉^𝑇𝑖𝑉_𝑗=0,1≤𝑗<𝑖≤𝑠.(A.1) We prove the conclusion by induction, and two steps are required.

Step 1. We will show that tr𝑅^𝑇𝑖+1𝑅_𝑖=0,tr𝑈^𝑇𝑖+1𝑈_𝑖+𝑉^𝑇𝑖+1𝑉_𝑖=0,𝑖=1,2,…,𝑠−1.(A.2)
To prove this conclusion, we also use induction.
For 𝑖=1, by Algorithm 2.1, we have that t=rtr𝑅⎛⎜⎜⎝^𝑇2𝑅𝑅₁₁−‖‖𝑅¹‖‖²‖‖𝑈₁‖‖²+‖‖𝑉₁‖‖²diagΦ𝑈₁,𝑉₁,Ψ𝑈₁,𝑉₁^𝑇𝑅₁=‖‖𝑅₁⎞⎟⎟⎠‖‖²−‖‖𝑅¹‖‖²‖‖𝑈₁‖‖²+‖‖𝑉₁‖‖²trdiagΦ𝑈₁,𝑉₁,Ψ𝑈₁,𝑉₁^𝑇=‖‖𝑅×diag𝑀−Φ𝑋₁,𝑌₁,𝑁−Ψ𝑋₁,𝑌₁₁‖‖²−‖‖𝑅¹‖‖²‖‖𝑈₁‖‖²+‖‖𝑉₁‖‖²=×‖‖𝑅trΦ𝑈₁,𝑉₁^𝑇𝑀−Φ𝑋₁,𝑌₁+Ψ𝑈₁,𝑉₁^𝑇𝑁−Ψ𝑋₁,𝑌₁₁‖‖²−‖‖𝑅¹‖‖²‖‖𝑈₁‖‖²+‖‖𝑉₁‖‖²tr𝑈^𝑇1𝐴^𝑇𝑀−Φ𝑋₁,𝑌₁𝐵^𝑇+𝑈^𝑇1𝐸^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐹^𝑇=‖‖𝑅₁−𝑉^𝑇1𝐶^𝑇𝑀−Φ𝑋₁,𝑌₁𝐷^𝑇−𝑉^𝑇1𝐺^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐻^𝑇‖‖²−‖‖𝑅¹‖‖²‖‖𝑈₁‖‖²+‖‖𝑉₁‖‖²×tr𝑈^𝑇1𝐴^𝑇𝑀−Φ𝑋₁,𝑌₁𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐹^𝑇+𝐴_𝑇2𝑀−Φ𝑋₁,𝑌₁𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐹^𝑇+𝑃𝐴_𝑇2𝑀−Φ𝑋₁,𝑌₁𝐵^𝑇𝑄+𝑃𝐸^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐹^𝑇𝑄−𝑃𝐴_𝑇2𝑀−Φ𝑋₁,𝑌₁𝐵^𝑇𝑄+𝑃𝐸^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐹^𝑇𝑄+𝑉_𝑇12−𝐶^𝑇𝑀−Φ𝑋₁,𝑌₁𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐻^𝑇+−𝐶_𝑇2𝑀−Φ𝑋₁,𝑌₁𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐻^𝑇+−𝑅𝐶_𝑇2𝑀−Φ𝑋₁,𝑌₁𝐷^𝑇𝑆−𝑅𝐺^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐻^𝑇𝑆−−𝑅𝐶_𝑇2𝑀−Φ𝑋₁,𝑌₁𝐷^𝑇𝑆−𝑅𝐺^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐻^𝑇𝑆=‖‖𝑅2₁‖‖²−‖‖𝑅¹‖‖²‖‖𝑈₁‖‖²+‖‖𝑉₁‖‖²×tr𝑈^𝑇1𝐴^𝑇𝑀−Φ𝑋₁,𝑌₁𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐹^𝑇+𝑃𝐴_𝑇2𝑀−Φ𝑋₁,𝑌₁𝐵^𝑇𝑄+𝑃𝐸^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐹^𝑇𝑄+𝑉_𝑇12−𝐶^𝑇𝑀−Φ𝑋₁,𝑌₁𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐻^𝑇+−𝑅𝐶_𝑇2𝑀−Φ𝑋₁,𝑌₁𝐷^𝑇𝑆−𝑅𝐺^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐻^𝑇𝑆=‖‖𝑅2₁‖‖²−‖‖𝑅¹‖‖²‖‖𝑈₁‖‖²+‖‖𝑉₁‖‖²tr𝑈_𝑇2tr𝑈^𝑇1𝑈₁+𝑉^𝑇1𝑉₁=0,=tr𝑈₁𝐴_𝑇+tr𝑉^𝑇2𝑉₁𝑀−Φ𝑋₂,𝑌₂𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋₂,𝑌₂𝐹^𝑇+𝑃𝐴_𝑇2𝑀−Φ𝑋₂,𝑌₂𝐵^𝑇𝑄+𝑃𝐸^𝑇𝑁−Ψ𝑋₂,𝑌₂𝐹^𝑇𝑄2+‖‖𝑅²‖‖²‖‖𝑅₁‖‖²𝑈₁^𝑇𝑈₁+tr−𝐶_𝑇⎞⎟⎟⎠𝑀−Φ𝑋₂,𝑌₂𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋₂,𝑌₂𝐻^𝑇+−𝑅𝐶_𝑇2𝑀−Φ𝑋₂,𝑌₂𝐷^𝑇𝑆−𝑅𝐺^𝑇𝑁−Ψ𝑋₂,𝑌₂𝐻^𝑇𝑆2+‖‖𝑅²‖‖²‖‖𝑅₁‖‖²𝑉₁^𝑇𝑉₁=tr⎛⎜⎜⎝𝐴_𝑇⎞⎟⎟⎠𝑀−Φ𝑋₂,𝑌₂𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋₂,𝑌₂𝐹^𝑇+‖‖𝑅²‖‖²‖‖𝑅₁‖‖²𝑈₁^𝑇𝑈₁+tr⎛⎜⎜⎝−𝐶_𝑇⎞⎟⎟⎠𝑀−Φ𝑋₂,𝑌₂𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋₂,𝑌₂𝐻^𝑇+‖‖𝑅²‖‖²‖‖𝑅₁‖‖²𝑉₁^𝑇𝑉₁=tr𝑈_𝑇1⎞⎟⎟⎠+𝑉_𝑇1𝐴^𝑇𝑀−Φ𝑋₂,𝑌₂𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋₂,𝑌₂𝐹^𝑇−𝐶^𝑇𝑀−Φ𝑋₂,𝑌₂𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋₂,𝑌₂𝐻^𝑇+‖‖𝑅²‖‖²‖‖𝑅₁‖‖²‖‖𝑈¹=tr𝑀−Φ𝑋₂‖‖²+‖‖𝑉₁‖‖²−𝑁−Ψ𝑋₂,𝑌₂^𝑇𝐴𝑈₁𝐵+𝑁−Ψ𝑋₂,𝑌₂^𝑇𝐸𝑈₁𝐹−𝑀−Φ𝑋₂,𝑌₂^𝑇𝐶𝑉₁𝐷,𝑌₂^𝑇𝐺𝑉₁𝐻+‖‖𝑅²‖‖²‖‖𝑅₁‖‖²‖‖𝑈¹=trdiag𝑀−Φ𝑋₂‖‖²+‖‖𝑉₁‖‖²+‖‖𝑅,𝑌₂^𝑇,𝑁−Ψ𝑋₂,𝑌₂^𝑇diagΦ𝑈¹,𝑉₁,Ψ𝑈₁,𝑉₁₂‖‖²‖‖𝑅₁‖‖²‖‖𝑈¹=‖‖𝑈₁‖‖²+‖‖𝑉₁‖‖²‖‖²+‖‖𝑉₁‖‖²‖‖𝑅₁‖‖²tr𝑅^𝑇2𝑅₁−𝑅₂+‖‖𝑅²‖‖²‖‖𝑅₁‖‖²‖‖𝑈¹‖‖²+‖‖𝑉₁‖‖²=0.(A.3)
Assume that (A.2) holds for 𝑖=𝑘−1, that is, tr(𝑅^𝑇𝑘𝑅_𝑘−1)=0,tr(𝑈^𝑇𝑘𝑈_𝑘−1+𝑉^𝑇𝑘𝑉_𝑘−1)=0.
When 𝑖=𝑘, we have that t=rtr𝑅⎛⎜⎜⎝^𝑇𝑘+1𝑅𝑅_𝑘𝑘−‖‖𝑅^𝑘‖‖²‖‖𝑈_𝑘‖‖²+‖‖𝑉_𝑘‖‖²diagΦ𝑈_𝑘,𝑉_𝑘,Ψ𝑈_𝑘,𝑉_𝑘^𝑇𝑅_𝑘=‖‖𝑅_𝑘⎞⎟⎟⎠‖‖²−‖‖𝑅^𝑘‖‖²‖‖𝑈_𝑘‖‖²+‖‖𝑉_𝑘‖‖²trdiagΦ𝑈_𝑘,𝑉_𝑘,Ψ𝑈_𝑘,𝑉_𝑘^𝑇=×‖‖𝑅diag𝑀−Φ𝑋_𝑘,𝑌_𝑘,𝑁−Ψ𝑋_𝑘,𝑌_𝑘_𝑘‖‖²−‖‖𝑅^𝑘‖‖²‖‖𝑈_𝑘‖‖²+‖‖𝑉_𝑘‖‖²=×‖‖𝑅trΦ𝑈_𝑘,𝑉_𝑘^𝑇𝑀−Φ𝑋_𝑘,𝑌_𝑘+Ψ𝑈_𝑘,𝑉_𝑘^𝑇𝑁−Ψ𝑋_𝑘,𝑌_𝑘_𝑘‖‖²−‖‖𝑅^𝑘‖‖²‖‖𝑈_𝑘‖‖²+‖‖𝑉_𝑘‖‖²tr𝑈^𝑇𝑘𝐴^𝑇𝑀−Φ𝑋_𝑘,𝑌_𝑘𝐵^𝑇+𝑈^𝑇𝑘𝐸^𝑇𝑁−Ψ𝑋_𝑘,𝑌_𝑘𝐹^𝑇=‖‖𝑅_𝑘−𝑉^𝑇𝑘𝐶^𝑇𝑀−Φ𝑋_𝑘,𝑌_𝑘𝐷^𝑇−𝑉^𝑇𝑘𝐺^𝑇𝑁−Ψ𝑋_𝑘,𝑌_𝑘𝐻^𝑇‖‖²−‖‖𝑅^𝑘‖‖²‖‖𝑈_𝑘‖‖²+‖‖𝑉_𝑘‖‖²tr𝑈^𝑇𝑘𝐴^𝑇𝑀−Φ𝑋_𝑘,𝑌_𝑘𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋_𝑘,𝑌_𝑘𝐹^𝑇+𝐴_𝑇2𝑀−Φ𝑋_𝑘,𝑌_𝑘𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋_𝑘,𝑌_𝑘𝐹^𝑇+𝑃𝐴_𝑇2𝑀−Φ𝑋_𝑘,𝑌_𝑘𝐵^𝑇𝑄+𝑃𝐸^𝑇𝑁−Ψ𝑋_𝑘,𝑌_𝑘𝐹^𝑇𝑄−𝑃𝐴_𝑇2𝑀−Φ𝑋_𝑘,𝑌_𝑘𝐵^𝑇𝑄+𝑃𝐸^𝑇𝑁−Ψ𝑋_𝑘,𝑌_𝑘𝐹^𝑇𝑄+𝑉_𝑇𝑘2−𝐶^𝑇𝑀−Φ𝑋_𝑘,𝑌_𝑘𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋_𝑘,𝑌_𝑘𝐻^𝑇+−𝐶_𝑇2𝑀−Φ𝑋_𝑘,𝑌_𝑘𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋_𝑘,𝑌_𝑘𝐻^𝑇+−𝑅𝐶_𝑇2𝑀−Φ𝑋_𝑘,𝑌_𝑘𝐷^𝑇𝑆−𝑅𝐺^𝑇𝑁−Ψ𝑋_𝑘,𝑌_𝑘𝐻^𝑇𝑆−−𝑅𝐶_𝑇2𝑀−Φ𝑋_𝑘,𝑌_𝑘𝐷^𝑇𝑆−𝑅𝐺^𝑇𝑁−Ψ𝑋_𝑘,𝑌_𝑘𝐻^𝑇𝑆=‖‖𝑅2_𝑘‖‖²−‖‖𝑅^𝑘‖‖²‖‖𝑈_𝑘‖‖²+‖‖𝑉_𝑘‖‖²tr𝑈^𝑇𝑘𝐴^𝑇𝑀−Φ𝑋_𝑘,𝑌_𝑘𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋_𝑘,𝑌_𝑘𝐹^𝑇+𝑃𝐴_𝑇2𝑀−Φ𝑋_𝑘,𝑌_𝑘𝐵^𝑇𝑄+𝑃𝐸^𝑇𝑁−Ψ𝑋_𝑘,𝑌_𝑘𝐹^𝑇𝑄+𝑉_𝑇𝑘2−𝐶^𝑇𝑀−Φ𝑋_𝑘,𝑌_𝑘𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋_𝑘,𝑌_𝑘𝐻^𝑇+−𝑅𝐶_𝑇2𝑀−Φ𝑋_𝑘,𝑌_𝑘𝐷^𝑇𝑆−𝑅𝐺^𝑇𝑁−Ψ𝑋_𝑘,𝑌_𝑘𝐻^𝑇𝑆=‖‖𝑅2_𝑘‖‖²−‖‖𝑅^𝑘‖‖²‖‖𝑈_𝑘‖‖²+‖‖𝑉_𝑘‖‖²tr𝑈_𝑇𝑘+1tr𝑈^𝑇𝑘𝑈_𝑘+𝑉^𝑇𝑘𝑉_𝑘=0,=tr𝑈𝐴_𝑘𝑇+tr𝑉^𝑇𝑘+1𝑉_𝑘𝑀−Φ𝑋_𝑘+1,𝑌_𝑘+1𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋_𝑘+1,𝑌_𝑘+1𝐹^𝑇+𝑃𝐴_𝑇2𝑀−Φ𝑋_𝑘+1,𝑌_𝑘+1𝐵^𝑇𝑄+𝑃𝐸^𝑇𝑁−Ψ𝑋_𝑘+1,𝑌_𝑘+1𝐹^𝑇𝑄+‖‖𝑅_𝑘+12‖‖²‖‖𝑅_𝑘‖‖²𝑈_𝑘^𝑇𝑈_𝑘+tr−𝐶_𝑇⎞⎟⎟⎠𝑀−Φ𝑋_𝑘+1,𝑌_𝑘+1𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋_𝑘+1,𝑌_𝑘+1𝐻^𝑇+−𝑅𝐶_𝑇2𝑀−Φ𝑋_𝑘+1,𝑌_𝑘+1𝐷^𝑇𝑆−𝑅𝐺^𝑇𝑁−Ψ𝑋_𝑘+1,𝑌_𝑘+1𝐻^𝑇𝑆+‖‖𝑅_𝑘+12‖‖²‖‖𝑅_𝑘‖‖²𝑉_𝑘^𝑇𝑉_𝑘=tr⎛⎜⎜⎝𝐴_𝑇⎞⎟⎟⎠𝑀−Φ𝑋_𝑘+1,𝑌_𝑘+1𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋_𝑘+1,𝑌_𝑘+1𝐹^𝑇+‖‖𝑅^𝑘+1‖‖²‖‖𝑅_𝑘‖‖²𝑈_𝑘^𝑇𝑈_𝑘+tr⎛⎜⎜⎝−𝐶_𝑇⎞⎟⎟⎠𝑀−Φ𝑋_𝑘+1,𝑌_𝑘+1𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋_𝑘+1,𝑌_𝑘+1𝐻^𝑇+‖‖𝑅^𝑘+1‖‖²‖‖𝑅_𝑘‖‖²𝑉_𝑘^𝑇𝑉_𝑘=tr𝑈_𝑇𝑘⎞⎟⎟⎠+𝑉_𝑇𝑘𝐴^𝑇𝑀−Φ𝑋_𝑘+1,𝑌_𝑘+1𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋_𝑘+1,𝑌_𝑘+1𝐹^𝑇+‖‖𝑅−𝐶^𝑇𝑀−Φ𝑋_𝑘+1,𝑌_𝑘+1𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋_𝑘+1,𝑌_𝑘+1𝐻^𝑇_𝑘+1‖‖²‖‖𝑅_𝑘‖‖²‖‖𝑈^𝑘=tr𝑀−Φ𝑋_𝑘+1‖‖²+‖‖𝑉_𝑘‖‖²−𝑀−Φ𝑋_𝑘+1,𝑌_𝑘+1^𝑇𝐴𝑈_𝑘𝐵+𝑁−Ψ𝑋_𝑘+1,𝑌_𝑘+1^𝑇𝐸𝑈_𝑘𝐹−𝑁−Ψ𝑋_𝑘+1,𝑌_𝑘+1^𝑇𝐶𝑉_𝑘𝐷,𝑌_𝑘+1^𝑇𝐺𝑉_𝑘𝐻+‖‖𝑅^𝑘+1‖‖²‖‖𝑅_𝑘‖‖²‖‖𝑈^𝑘=trdiag𝑀−Φ𝑋_𝑘+1‖‖²+‖‖𝑉_𝑘‖‖²+‖‖𝑅,𝑌_𝑘+1^𝑇,𝑁−Ψ𝑋_𝑘+1,𝑌_𝑘+1^𝑇diagΦ𝑈^𝑘,𝑉_𝑘,Ψ𝑈_𝑘,𝑉_𝑘_𝑘+1‖‖²‖‖𝑅_𝑘‖‖²‖‖𝑈^𝑘=‖‖𝑈_𝑘‖‖²+‖‖𝑉_𝑘‖‖²‖‖²+‖‖𝑉_𝑘‖‖²‖‖𝑅_𝑘‖‖²tr𝑅^𝑇𝑘+1𝑅_𝑘−𝑅_𝑘+1+‖‖𝑅^𝑘+1‖‖²‖‖𝑅_𝑘‖‖²‖‖𝑈^𝑘‖‖²+‖‖𝑉_𝑘‖‖²=0.(A.4) Hence, (A.2) holds for 𝑖=𝑘. Therefore, (A.2) holds by the principle of induction.

Step 2. We show that tr𝑅^𝑇𝑖+1𝑅_𝑗=0,tr𝑈^𝑇𝑖+1𝑈_𝑗+𝑉^𝑇𝑖+1𝑉_𝑗=0,𝑗=1,2,…,𝑖,∀𝑖≥1.(A.5)
When 𝑖=1, (A.5) holds.
Assume that tr𝑅^𝑇𝑖𝑅_𝑗=0,tr𝑈^𝑇𝑖𝑈_𝑗+𝑉^𝑇𝑖𝑉_𝑗=0,𝑗=1,2,…,𝑠−1,∀𝑠≥2,(A.6) then we show that tr𝑅^𝑇𝑖+1𝑅_𝑗=0,tr𝑈^𝑇𝑖+1𝑈_𝑗+𝑉^𝑇𝑖+1𝑉_𝑗=0,𝑗=1,2,…,𝑠.(A.7)
In fact, we have that t=rtr𝑅⎛⎜⎜⎝^𝑇𝑖+1𝑅𝑅_𝑗𝑖−‖‖𝑅^𝑖‖‖²‖‖𝑈_𝑖‖‖²+‖‖𝑉_𝑖‖‖²diagΦ𝑈_𝑖,𝑉_𝑖,Ψ𝑈_𝑖,𝑉_𝑖^𝑇𝑅_𝑗=tr𝑅_𝑇𝑖⎞⎟⎟⎠𝑅_𝑗−‖‖𝑅^𝑖‖‖²‖‖𝑈_𝑖‖‖²+‖‖𝑉_𝑖‖‖²trdiagΦ𝑈_𝑖,𝑉_𝑖,Ψ𝑈_𝑖,𝑉_𝑖^𝑇=×−diag‖‖𝑅𝑀_𝑖−Φ𝑋_𝑗,𝑌_𝑗,𝑁−Ψ𝑋_𝑗,𝑌_𝑗‖‖²‖‖𝑈_𝑖‖‖²+‖‖𝑉_𝑖‖‖²=−‖‖𝑅_𝑖trΦ𝑈_𝑖,𝑉_𝑖^𝑇𝑀−Φ𝑋_𝑗,𝑌_𝑗+Ψ𝑈_𝑖,𝑉_𝑖^𝑇𝑁−Ψ𝑋_𝑗,𝑌_𝑗‖‖²‖‖𝑈_𝑖‖‖²+‖‖𝑉_𝑖‖‖²tr𝑈^𝑇𝑖𝐴^𝑇𝑀−Φ𝑋_𝑗,𝑌_𝑗𝐵^𝑇+𝑈^𝑇𝑖𝐸^𝑇𝑁−Ψ𝑋_𝑗,𝑌_𝑗𝐹^𝑇=−‖‖𝑅_𝑖−𝑉^𝑇𝑖𝐶^𝑇𝑀−Φ𝑋_𝑗,𝑌_𝑗𝐷^𝑇−𝑉^𝑇𝑖𝐺^𝑇𝑁−Ψ𝑋_𝑗,𝑌_𝑗𝐻^𝑇‖‖²‖‖𝑈_𝑖‖‖²+‖‖𝑉_𝑖‖‖²tr𝑈^𝑇𝑖𝐴^𝑇𝑀−Φ𝑋_𝑗,𝑌_𝑗𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋_𝑗,𝑌_𝑗𝐹^𝑇+𝐴_𝑇2𝑀−Φ𝑋_𝑗,𝑌_𝑗𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋_𝑗,𝑌_𝑗𝐹^𝑇+𝑃𝐴_𝑇2𝑀−Φ𝑋_𝑗,𝑌_𝑗𝐵^𝑇𝑄+𝑃𝐸^𝑇𝑁−Ψ𝑋_𝑗,𝑌_𝑗𝐹^𝑇𝑄−𝑃𝐴_𝑇2𝑀−Φ𝑋_𝑗,𝑌_𝑗𝐵^𝑇𝑄+𝑃𝐸^𝑇𝑁−Ψ𝑋_𝑗,𝑌_𝑗𝐹^𝑇𝑄+𝑉_𝑇𝑖2−𝐶^𝑇𝑀−Φ𝑋_𝑗,𝑌_𝑗𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋_𝑗,𝑌_𝑗𝐻^𝑇+−𝐶_𝑇2𝑀−Φ𝑋_𝑗,𝑌_𝑗𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋_𝑗,𝑌_𝑗𝐻^𝑇+−𝑅𝐶_𝑇2𝑀−Φ𝑋_𝑗,𝑌_𝑗𝐷^𝑇𝑆−𝑅𝐺^𝑇𝑁−Ψ𝑋_𝑗,𝑌_𝑗𝐻^𝑇𝑆−−𝑅𝐶_𝑇2𝑀−Φ𝑋_𝑗,𝑌_𝑗𝐷^𝑇𝑆−𝑅𝐺^𝑇𝑁−Ψ𝑋_𝑗,𝑌_𝑗𝐻^𝑇𝑆=−‖‖𝑅_𝑖2‖‖²‖‖𝑈_𝑖‖‖²+‖‖𝑉_𝑖‖‖²tr𝑈^𝑇𝑖𝐴^𝑇𝑀−Φ𝑋_𝑗,𝑌_𝑗𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋_𝑗,𝑌_𝑗𝐹^𝑇+𝑃𝐴_𝑇2𝑀−Φ𝑋_𝑗,𝑌_𝑗𝐵^𝑇𝑄+𝑃𝐸^𝑇𝑁−Ψ𝑋_𝑗,𝑌_𝑗𝐹^𝑇𝑄+𝑉_𝑇𝑖2−𝐶^𝑇𝑀−Φ𝑋_𝑗,𝑌_𝑗𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋_𝑗,𝑌_𝑗𝐻^𝑇+−𝑅𝐶_𝑇2𝑀−Φ𝑋_𝑗,𝑌_𝑗𝐷^𝑇𝑆−𝑅𝐺^𝑇𝑁−Ψ𝑋_𝑗,𝑌_𝑗𝐻^𝑇𝑆=−‖‖𝑅_𝑖2‖‖²‖‖𝑈_𝑖‖‖²+‖‖𝑉_𝑖‖‖²tr𝑈^𝑇𝑖𝑈^𝑗−‖‖𝑅^𝑗‖‖²‖‖𝑅_𝑗−1‖‖²𝑈_𝑗−1+𝑉^𝑇𝑖𝑉^𝑗−‖‖𝑅^𝑗‖‖²‖‖𝑅_𝑗−1‖‖²𝑉_𝑗−1=−‖‖𝑅_𝑖‖‖²‖‖𝑈_𝑖‖‖²+‖‖𝑉_𝑖‖‖²tr𝑈^𝑇𝑖𝑈_𝑗+𝑉^𝑇𝑖𝑉_𝑗+‖‖𝑅^𝑖‖‖²‖‖𝑅_𝑗‖‖²‖‖𝑈^𝑖‖‖²+‖‖𝑉_𝑖‖‖²‖‖𝑅^𝑗−1‖‖⁴×tr𝑈^𝑇𝑖𝑈_𝑗−1+tr𝑉^𝑇𝑖𝑉_𝑗−1=0.(A.8) From the above results, we have tr(𝑅^𝑇𝑖+1𝑅_𝑗+1)=0,𝑗=1,2,…,𝑠−1, and t=rtr𝑈⎛⎜⎜⎝⎡⎢⎢⎣^𝑇𝑖+1𝐴𝑈_𝑗𝑇+tr𝑉^𝑇𝑖+1𝑉_𝑗𝑀−Φ𝑋_𝑖+1,𝑌_𝑖+1𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋_𝑖+1,𝑌_𝑖+1𝐹^𝑇+𝑃𝐴_𝑇2𝑀−Φ𝑋_𝑖+1,𝑌_𝑖+1𝐵^𝑇𝑄+𝑃𝐸^𝑇𝑁−Ψ𝑋_𝑖+1,𝑌_𝑖+1𝐹^𝑇𝑄+‖‖𝑅_𝑖+12‖‖²‖‖𝑅_𝑖‖‖²𝑈_𝑖^𝑇𝑈_𝑗+tr⎛⎜⎜⎝−𝐶_𝑇⎞⎟⎟⎠𝑀−Φ𝑋_𝑖+1,𝑌_𝑖+1𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋_𝑖+1,𝑌_𝑖+1𝐻^𝑇+−𝑅𝐶_𝑇2𝑀−Φ𝑋_𝑖+1,𝑌_𝑖+1𝐷^𝑇𝑆−𝑅𝐺^𝑇𝑁−Ψ𝑋_𝑖+1,𝑌_𝑖+1𝐻^𝑇𝑆+‖‖𝑅_𝑖+12‖‖²‖‖𝑅_𝑖‖‖²𝑉_𝑖^𝑇𝑉_𝑗=tr⎛⎜⎜⎝𝐴_𝑇⎞⎟⎟⎠𝑀−Φ𝑋_𝑖+1,𝑌_𝑖+1𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋_𝑖+1,𝑌_𝑖+1𝐹^𝑇+‖‖𝑅^𝑖+1‖‖²‖‖𝑅_𝑖‖‖²𝑈_𝑖^𝑇𝑈_𝑗+tr⎛⎜⎜⎝−𝐶_𝑇⎞⎟⎟⎠𝑀−Φ𝑋_𝑖+1,𝑌_𝑖+1𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋_𝑖+1,𝑌_𝑖+1𝐻^𝑇+‖‖𝑅^𝑖+1‖‖²‖‖𝑅_𝑖‖‖²𝑉_𝑖^𝑇𝑉_𝑗=tr𝑈_𝑇𝑗⎞⎟⎟⎠+𝑉_𝑇𝑗𝐴^𝑇𝑀−Φ𝑋_𝑖+1,𝑌_𝑖+1𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋_𝑖+1,𝑌_𝑖+1𝐹^𝑇+‖‖𝑅−𝐶^𝑇𝑀−Φ𝑋_𝑖+1,𝑌_𝑖+1𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋_𝑖+1,𝑌_𝑖+1𝐻^𝑇_𝑖+1‖‖²‖‖𝑅_𝑖‖‖²=tr𝑀−Φtr𝑋𝑈^𝑇𝑖_𝑖+1𝑈_𝑗+tr𝑉^𝑇𝑖𝑉_𝑗−𝑀−Φ𝑋,_𝑖+1𝑌_𝑖+1^𝑇𝐴𝑈_𝑗𝐵+𝑁−Ψ𝑋_𝑖+1,𝑌_𝑖+1^𝑇𝐸𝑈_𝑗𝐹−𝑁−Ψ𝑋_𝑖+1,𝑌_𝑖+1^𝑇𝐶𝑉_𝑗𝐷,𝑌_𝑖+1^𝑇𝐺𝑉_𝑗𝐻+‖‖𝑅^𝑖+1‖‖²‖‖𝑅_𝑖‖‖²=trdiag𝑀−Φ𝑋_𝑖+1tr𝑈^𝑇𝑖𝑈_𝑗+tr𝑉^𝑇𝑖𝑉_𝑗+‖‖𝑅,𝑌_𝑖+1^𝑇,𝑁−Ψ𝑋_𝑖+1,𝑌_𝑖+1^𝑇diagΦ𝑈^𝑗,𝑉_𝑗,Ψ𝑈_𝑗,𝑉_𝑗_𝑖+1‖‖²‖‖𝑅_𝑖‖‖²=‖‖𝑈tr𝑈^𝑇𝑖𝑈_𝑗+tr𝑉^𝑇𝑖𝑉_𝑗_𝑗‖‖²+‖‖𝑉_𝑗‖‖²‖‖𝑅_𝑗‖‖²tr𝑅^𝑇𝑖+1𝑅_𝑗−𝑅_𝑗+1+‖‖𝑅^𝑖+1‖‖²‖‖𝑅_𝑖‖‖²tr𝑈^𝑇𝑖𝑈_𝑗+tr𝑉^𝑇𝑖𝑉_𝑗=0.(A.9)By the principle of induction, (A.5) holds.
Noting that (A.1) is implied in Steps 1 and 2 by the principle of induction. This completes the proof.

B. The Proof of Lemma 2.3

We proof the conclusion by induction.

For 𝑖=1, we have thatt=rtr𝑋^∗−𝑋𝑋_1∗^𝑇𝑈₁+𝑌^∗−𝑌₁^𝑇𝑉₁−𝑋₁^𝑇𝐴^𝑇𝑀−Φ𝑋₁,𝑌₁𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐹^𝑇+𝑃𝐴_𝑇2𝑀−Φ𝑋₁,𝑌₁𝐵^𝑇𝑄+𝑃𝐸^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐹^𝑇𝑄+𝑌_∗2−𝑌₁^𝑇−𝐶^𝑇𝑀−Φ𝑋₁,𝑌₁𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐻^𝑇+−𝑅𝐶_𝑇2𝑀−Φ𝑋₁,𝑌₁𝐷^𝑇𝑆−𝑅𝐺^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐻^𝑇𝑆=tr𝑋_∗2+𝑌_∗−𝑋₁^𝑇𝐴^𝑇𝑀−Φ𝑋₁,𝑌₁𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐹^𝑇=tr𝑀−−Φ𝑌₁𝑋^𝑇₁−𝐶^𝑇𝑀−Φ𝑋₁,𝑌₁𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋₁,𝑌₁𝐻^𝑇−𝑀−Φ𝑋,₁𝑌₁^𝑇𝐴𝑋^∗−𝑋₁𝐵+𝑁−Ψ𝑋₁,𝑌₁^𝑇𝐸𝑋^∗−𝑋₁𝐹=tr⎛⎜⎜⎝⎛⎜⎜⎝𝑀−Φ𝑋₁,𝑌₁^𝑇𝐶𝑍^∗−𝑍₁𝐷−𝑁−Ψ𝑋₁,𝑌₁^𝑇𝐺𝑌^∗−𝑌₁𝐻0,𝑌₁^𝑇𝑁−Ψ0𝑋₁,𝑌₁^𝑇⎛⎜⎜⎝𝐴𝑋_∗⎞⎟⎟⎠−𝑋₁𝐵0−𝐶𝑌^∗−𝑌₁𝐷𝐸𝑋_∗0=tr⎛⎜⎜⎜⎝⎛⎜⎜⎝𝑀−Φ𝑋₁−𝑋₁𝐹−𝐺𝑌^∗−𝑌₁𝐻⎞⎟⎟⎠⎞⎟⎟⎠0,𝑌₁𝑁−Ψ0𝑋₁,𝑌₁⎞⎟⎟⎠^𝑇⎛⎜⎜⎝𝑀−Φ0𝑋₁,𝑌₁𝑁−Ψ0𝑋₁=‖‖𝑅₁,𝑌₁⎞⎟⎟⎠⎞⎟⎟⎟⎠‖‖².(B.1)

Assume that (2.5) holds for 𝑖=𝑘. When 𝑖=𝑘+1, by Algorithm 2.1, we have that t=rtr𝑋^∗𝑋−_∗𝑋_𝑘+1^𝑇𝑈_𝑘+1+𝑌^∗−𝑌_𝑘+1^𝑇𝑉_𝑘+1−𝑋_𝑘+1^𝑇𝐴^𝑇𝑀−Φ𝑋_𝑘+1,𝑌_𝑘+1𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋_𝑘+1,𝑌_𝑘+1𝐹^𝑇+𝑃𝐴_𝑇2𝑀−Φ𝑋_𝑘+1,𝑌_𝑘+1𝐵^𝑇𝑄+𝑃𝐸^𝑇𝑁−Ψ𝑋_𝑘+1,𝑌_𝑘+1𝐹^𝑇𝑄+‖‖𝑅_𝑘+12‖‖²‖‖𝑅_𝑘‖‖²𝑈_𝑘+𝑌_∗−𝑌_𝑘+1^𝑇−𝐶^𝑇𝑀−Φ𝑋_𝑘+1,𝑌_𝑘+1𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋_𝑘+1,𝑌_𝑘+1𝐻^𝑇+−𝑅𝐶_𝑇2𝑀−Φ𝑋_𝑘+1,𝑌_𝑘+1𝐷^𝑇𝑆−𝑅𝐺^𝑇𝑁−Ψ𝑋_𝑘+1,𝑌_𝑘+1𝐻^𝑇𝑆+‖‖𝑅_𝑘+12‖‖²‖‖𝑅_𝑘‖‖²𝑉_𝑘=tr𝑋_∗+𝑌_∗−𝑋_𝑘+1^𝑇𝐴^𝑇𝑀−Φ𝑋_𝑘+1,𝑌_𝑘+1𝐵^𝑇+𝐸^𝑇𝑁−Ψ𝑋_𝑘+1,𝑌_𝑘+1𝐹^𝑇+‖‖𝑅−_𝑘+1𝑌_𝑘+1^𝑇−𝐶^𝑇𝑀−Φ𝑋_𝑘+1,𝑌_𝑘+1𝐷^𝑇−𝐺^𝑇𝑁−Ψ𝑋_𝑘+1,𝑌_𝑘+1𝐻^𝑇‖‖²‖‖𝑅_𝑘‖‖²=tr𝑀−Φtr𝑋𝑋^∗_𝑘+1−𝑋_𝑘+1^𝑇𝑈_𝑘+𝑌^∗−𝑌_𝑘+1^𝑇𝑉_𝑘−𝑀−Φ𝑋,_𝑘+1𝑌_𝑘+1^𝑇𝐴𝑋^∗−𝑋_𝑘+1𝐵+𝑁−Ψ𝑋_𝑘+1,𝑌_𝑘+1^𝑇𝐸𝑋^∗−𝑋_𝑘+1𝐹+‖‖𝑅_𝑘+1,𝑌_𝑘+1^𝑇𝐶𝑍^∗−𝑍_𝑘+1𝐷−𝑁−Ψ𝑋_𝑘+1,𝑌_𝑘+1^𝑇𝐺𝑌^∗−𝑌_𝑘+1𝐻‖‖²‖‖𝑅_𝑘‖‖²=tr⎛⎜⎜⎝⎛⎜⎜⎝𝑀−Φtr𝑋𝑋^∗_𝑘+1−𝑋_𝑘+1^𝑇𝑈_𝑘+𝑌^∗−𝑌_𝑘+1^𝑇𝑉_𝑘0,𝑌_𝑘+1^𝑇𝑁−Ψ𝑋0_𝑘+1,𝑌_𝑘+1^𝑇⎛⎜⎜⎝𝐴𝑋_∗⎞⎟⎟⎠−𝑋_𝑘+1𝐵0−𝐶𝑌^∗−𝑌_𝑘+1𝐷𝐸𝑋_∗0+‖‖𝑅_𝑘+1−𝑋_𝑘+1𝐹−𝐺𝑌^∗−𝑌_𝑘+1𝐻⎞⎟⎟⎠⎞⎟⎟⎠‖‖²‖‖𝑅_𝑘‖‖²=tr⎛⎜⎜⎝⎛⎜⎜⎝𝑀−Φtr𝑋𝑋^∗_𝑘+1−𝑋_𝑘+1^𝑇𝑈_𝑘+𝑌^∗−𝑌_𝑘+1^𝑇𝑉_𝑘0,𝑌_𝑘+1^𝑇𝑁−Ψ𝑋0_𝑘+1,𝑌_𝑘+1^𝑇⎛⎜⎜⎝𝑀−Φ𝑋_𝑘+1⎞⎟⎟⎠0,𝑌_𝑘+1𝑁−Ψ𝑋0_𝑘+1+‖‖𝑅_𝑘+1,𝑌_𝑘+1⎞⎟⎟⎠⎞⎟⎟⎠‖‖²‖‖𝑅_𝑘‖‖²=‖‖𝑅_𝑘+1tr𝑋^∗−𝑋_𝑘+1^𝑇𝑈_𝑘+𝑌^∗−𝑌_𝑘+1^𝑇𝑉_𝑘‖‖²+‖‖𝑅^𝑘+1‖‖²‖‖𝑅_𝑘‖‖²−‖‖𝑅_𝑘+1tr𝑋^∗−𝑋_𝑘^𝑇𝑈_𝑘+𝑌^∗−𝑌_𝑘^𝑇𝑉_𝑘‖‖²‖‖𝑈_𝑘‖‖²+‖‖𝑉_𝑘‖‖²tr𝑈^𝑇𝑘𝑈_𝑘+𝑉^𝑇𝑘𝑉_𝑘=‖‖𝑅_𝑘+1‖‖².(B.2)

Therefore, (2.5) holds for 𝑖=𝑘+1. Thus, (2.5) holds by the principal of induction. This completes the proof.

Acknowledgments

The authors are very much indebted to the anonymous referees and their editors for their constructive and valuable comments and suggestions which greatly improved the original copy of this paper. Grateful acknowledgements are given to Professor. Alain Miranville for his comments and suggestions that helped improve the second version of this paper greatly. This work was partially supported by the Research Fund Project (Natural Science 2010XJKYL018) and Natural Science Foundation of Sichuan Education Department (12ZB289) respectively. This work is also supported by Open Fund of Geomathematics Key Laboratory of Sichuan Province (scsxdz2011005) and Key Natural Science Foundation of Sichuan Education Department (12ZA008).