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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 152805, 28 pages
http://dx.doi.org/10.1155/2012/152805
Research Article

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

1School of Science, Sichuan University of Science and Engineering, Zigong 643000, China
2College of Management Science, Geomathematics Key Laboratory of Sichuan Province, Chengdu University of Technology, Chengdu 610059, China

Received 14 November 2011; Revised 13 March 2012; Accepted 25 March 2012

Academic Editor: AlainΒ Miranville

Copyright Β© 2012 Feng Yin and Guang-Xin Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations (π΄π‘‹π΅βˆ’πΆπ‘Œπ·,πΈπ‘‹πΉβˆ’πΊπ‘Œπ»)=(𝑀,𝑁), which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matrices 𝑋 and π‘Œ. When the matrix equations are consistent, for any initial generalized reflexive matrix pair [𝑋1,π‘Œ1], 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,π‘Œ] to a given matrix pair [𝑋0,π‘Œ0] in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair [ξ‚π‘‹βˆ—,ξ‚π‘Œβˆ—] of a new corresponding generalized coupled Sylvester matrix equation pair (π΄π‘‹π΅βˆ’πΆπ‘Œπ·,πΈπ‘‹πΉβˆ’πΊπ‘Œπ»)=(𝑀,𝑁), where 𝑀=π‘€βˆ’π΄π‘‹0𝐡+πΆπ‘Œ0𝐷,𝑁=π‘βˆ’πΈπ‘‹0𝐹+πΊπ‘Œ0𝐻. 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 β„›π‘šΓ—π‘› denote the set of all π‘šΓ—π‘› real matrices. We denote by the superscript 𝑇 the transpose of a matrix. In matrix space β„›π‘šΓ—π‘›, define inner product as ⟨𝐴,𝐡⟩=tr(𝐡𝑇𝐴) for all 𝐴,π΅βˆˆβ„›π‘šΓ—π‘›, where tr(𝐴) denotes the trace of a matrix 𝐴. ‖𝐴‖ represents the Frobenius norm of 𝐴. β„›(𝐴) represents the column space of 𝐴. vec(β‹…) represents the vector operator, that is, vec(𝐴)=(πšπ‘‡1,πšπ‘‡2,…,πšπ‘‡π‘›)π‘‡βˆˆβ„›π‘šπ‘› for the matrix 𝐴=(𝐚1,𝐚2,…,πšπ‘›)βˆˆβ„›π‘šΓ—π‘›,πšπ‘–βˆˆπ‘…π‘š,𝑖=1,2,…,𝑛. π΄βŠ—π΅ stands for the Kronecker product of matrices 𝐴 and 𝐡, diag(𝐴,𝐡) denotes the block diagonal matrix with 𝐴 and 𝐡 and being the main diagonal elements orderly. 𝐼𝑛 denotes the n-order identity matrix.

Definition 1.1 (see [1, 2]). A matrix π‘ƒβˆˆβ„›π‘›Γ—π‘› is said to be a generalized reflection matrix if 𝑃 satisfies that 𝑃𝑇=𝑃,𝑃2=𝐼.

Definition 1.2 (see [1, 2]). Let π‘ƒβˆˆβ„›π‘›Γ—π‘› and π‘„βˆˆβ„›π‘›Γ—π‘› be two generalized reflection matrices. A matrix π΄βˆˆβ„›π‘›Γ—π‘› is called generalized reflexive (or generalized antireflexive) with respect to the matrix pair (𝑃,𝑄) if 𝑃𝐴𝑄=𝐴(or𝑃𝐴𝑄=βˆ’π΄). The set of all n-by-n generalized reflexive matrices with respect to matrix pair (𝑃,𝑄) is denoted by β„›π‘Ÿπ‘›Γ—π‘›(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 𝑋𝑇=𝑋, then a reflexive matrix 𝑋 is called a generalized bisymmetric matrix, which has been studied in [12, 13]. Moreover, let 𝑃=𝑄=𝐽𝑛, 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. KΓ₯gstrΓΆm and Westin [18] developed a generalized Schur method by applying the QZ algorithm to solve (π΄π‘‹π΅βˆ’πΆπ‘Œπ·,πΈπ‘‹πΉβˆ’πΊπ‘Œπ»)=(𝑀,𝑁). Ding and Chen [19] presented an iterative least squares solutions of (π΄π‘‹π΅βˆ’πΆπ‘Œπ·,πΈπ‘‹πΉβˆ’πΊπ‘Œπ»)=(𝑀,𝑁) based on a hierarchical identification principle [20], in addition, by applying the hierarchical identification principle, KΔ±lΔ±Γ§man 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=𝐢1,𝐴3𝑋𝐡3=𝐢3. 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+π‘Œπ΄=𝐢,𝐴𝑋𝐴𝑇+π΅π‘Œπ΅π‘‡=𝐢, and (𝐴𝑇𝑋𝐴,𝐡𝑇𝑋𝐡)=(𝐢,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 βˆ‘π‘π‘—=1𝐴𝑖𝑗𝑋𝑗𝐡𝑖𝑗=𝑀𝑖,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 KΓ₯gstrΓΆm [39] proposed recursive block algorithms for solving the one-sided and coupled Sylvester matrix equations (π΄π‘‹βˆ’π‘Œπ΅,π·π‘‹βˆ’π‘ŒπΈ)=(𝐢,𝐹). Jonsson and KΓ₯gstrΓΆm [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 (π΄π‘Œβˆ’π‘π΅,πΆπ‘Œβˆ’π‘π·)=(𝐸,𝐹). Very recently, Dehghan and Hajarian [12] constructed an iterative algorithm to solve the generalized coupled Sylvester matrix equations (𝐴𝑋𝐡+πΆπ‘Œπ·,𝐸𝑋𝐹+πΊπ‘Œπ»)=(𝑀,𝑁) over generalized bisymmetric matrices. Huang et al. [13] present an iterative algorithm for the generalized coupled Sylvester matrix equations (π΄π‘Œβˆ’π‘π΅,πΆπ‘Œβˆ’π‘π·)=(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 (π΄π‘‹π΅βˆ’πΆπ‘Œπ·,πΈπ‘‹πΉβˆ’πΊπ‘Œπ»)=(𝑀,𝑁) and the optimal approximation problem over reflexive matrices. However, the generalized coupled Sylvester matrix equations (π΄π‘‹π΅βˆ’πΆπ‘Œπ·,πΈπ‘‹πΉβˆ’πΊπ‘Œπ»)=(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 π‘ƒβˆˆβ„›π‘šΓ—π‘š,π‘„βˆˆβ„›π‘›Γ—π‘›,π‘…βˆˆβ„›π‘ Γ—π‘ , and π‘†βˆˆβ„›π‘‘Γ—π‘‘ be generalized reflection matrices. For given matrices π΄βˆˆβ„›π‘Γ—π‘š,β€‰π΅βˆˆβ„›π‘›Γ—π‘ž,β€‰πΆβˆˆβ„›π‘Γ—π‘ ,β€‰π·βˆˆβ„›π‘‘Γ—π‘ž,π‘€βˆˆβ„›π‘Γ—π‘ž,β€‰πΈβˆˆβ„›π‘˜Γ—π‘š,β€‰πΉβˆˆβ„›π‘›Γ—π‘™,β€‰πΊβˆˆβ„›π‘˜Γ—π‘ ,β€‰π»βˆˆβ„›π‘‘Γ—π‘™,β€‰π‘βˆˆβ„›π‘˜Γ—π‘™, find a pair of matrices π‘‹βˆˆβ„›π‘Ÿπ‘šΓ—π‘›(𝑃,𝑄), π‘Œβˆˆβ„›π‘Ÿπ‘ Γ—π‘‘(𝑅,𝑆) such that π΄π‘‹π΅βˆ’πΆπ‘Œπ·=𝑀,πΈπ‘‹πΉβˆ’πΊπ‘Œπ»=𝑁.(1.1)

Problem 2. When Problem 1 is consistent, let 𝑆𝐸 denote the set of the generalized reflexive solutions of Problem 1, that is, 𝑆𝐸=ξ€½[]𝑋,π‘Œβˆ£π΄π‘‹π΅βˆ’πΆπ‘Œπ·=𝑀,πΈπ‘‹πΉβˆ’πΊπ‘Œπ»=𝑁,π‘Œβˆˆβ„›π‘Ÿπ‘šΓ—π‘›(𝑃,𝑄),π‘βˆˆβ„›π‘Ÿπ‘ Γ—π‘‘ξ€Ύ(𝑅,𝑆).(1.2) For a given matrix pair [π‘Œ0,𝑍0]βˆˆβ„›π‘Ÿπ‘šΓ—π‘›(𝑃,𝑄)Γ—β„›π‘Ÿπ‘ Γ—π‘‘(𝑅,𝑆), find [ξξπ‘Œ,𝑍]βˆˆπ‘†πΈ such that β€–β€–ξπ‘Œβˆ’π‘Œ0β€–β€–2+β€–β€–ξπ‘βˆ’π‘0β€–β€–2=min[]π‘Œ,π‘βˆˆπ‘†πΈξ‚†β€–β€–π‘Œβˆ’π‘Œ0β€–β€–2+β€–β€–π‘βˆ’π‘0β€–β€–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 (π΄π‘‹βˆ’π‘Œπ΅,πΆπ‘‹βˆ’π‘Œπ·)=(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 [π‘Œ1,𝑍1]βˆˆβ„›π‘Ÿπ‘šΓ—π‘›(𝑃,𝑄)Γ—β„›π‘Ÿπ‘ Γ—π‘‘(𝑅,𝑆), we can obtain a solution pair [π‘Œβˆ—,π‘βˆ—] of Problem 1 within finite iterative steps in the absence of round-off errors. Let 𝑋1=𝐴𝑇𝐾𝐡𝑇+𝐸𝑇𝐿𝐹𝑇+𝑃𝐴𝑇𝐾𝐡𝑇𝑄+𝑃𝐸𝑇𝐿𝐹𝑇𝑄 and π‘Œ1=βˆ’πΆπ‘‡πΎπ·π‘‡βˆ’πΊπ‘‡πΏπ»π‘‡βˆ’π‘…πΆπ‘‡πΎπ·π‘‡π‘†βˆ’π‘…πΊπ‘‡πΏπ»π‘‡π‘†, where πΎβˆˆβ„›π‘Γ—π‘ž, πΏβˆˆβ„›π‘˜Γ—π‘™ are arbitrary matrices, or more especially, let 𝑋1=0 and π‘Œ1=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:ΨΦ(𝑋,π‘Œ)=π΄π‘‹π΅βˆ’πΆπ‘Œπ·,(𝑋,π‘Œ)=πΈπ‘‹πΉβˆ’πΊπ‘Œπ».(2.1)

Algorithm 2.1. We have the following steps.

Step 1. Input matrices π΄βˆˆβ„›π‘Γ—π‘š,β€‰π΅βˆˆβ„›π‘›Γ—π‘ž,β€‰πΆβˆˆβ„›π‘Γ—π‘ ,β€‰π·βˆˆβ„›π‘‘Γ—π‘ž,β€‰π‘€βˆˆβ„›π‘Γ—π‘ž, πΈβˆˆβ„›π‘˜Γ—π‘š,β€‰πΉβˆˆβ„›π‘›Γ—π‘™,β€‰πΊβˆˆβ„›π‘˜Γ—π‘ ,β€‰π»βˆˆβ„›π‘‘Γ—π‘™,β€‰π‘βˆˆβ„›π‘˜Γ—π‘™, and four generalized reflection matrices π‘ƒβˆˆβ„›π‘šΓ—π‘š,β€‰π‘„βˆˆβ„›π‘›Γ—π‘›, π‘…βˆˆβ„›π‘ Γ—π‘ ,π‘†βˆˆβ„›π‘‘Γ—π‘‘.

Step 2. Choose two arbitrary matrices 𝑋1βˆˆβ„›π‘Ÿπ‘šΓ—π‘›(𝑃,𝑄), π‘Œ1βˆˆβ„›π‘Ÿπ‘ Γ—π‘‘(𝑅,𝑆). Compute 𝑅1𝑋=diagπ‘€βˆ’Ξ¦1,π‘Œ1𝑋,π‘βˆ’Ξ¨1,π‘Œ1,π‘ˆξ€Έξ€Έ1=12ξ€Ίπ΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦1,π‘Œ1𝐡𝑇+πΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨1,π‘Œ1𝐹𝑇+π‘ƒπ΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦1,π‘Œ1𝐡𝑇𝑄+π‘ƒπΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨1,π‘Œ1𝐹𝑇𝑄,𝑉1=12ξ€Ίβˆ’πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦1,π‘Œ1π·ξ€Έξ€Έπ‘‡βˆ’πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨1,π‘Œ1π»ξ€Έξ€Έπ‘‡βˆ’π‘…πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦1,π‘Œ1π·ξ€Έξ€Έπ‘‡π‘†βˆ’π‘…πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨1,π‘Œ1𝐻𝑇𝑆,π‘˜βˆΆ=1.(2.2)

Step 3. If π‘…π‘˜=0, then stop and [π‘‹π‘˜,π‘Œπ‘˜] is the solution of the generalized coupled Sylvester matrix equation (1.1); else if π‘…π‘˜β‰ 0, but π‘ˆπ‘˜=0 and π‘‰π‘˜=0, then stop and the generalized coupled Sylvester matrix equations (1.1) are not consistent over generalized reflexive matrices; else π‘˜βˆΆ=π‘˜+1.

Step 4. Compute π‘‹π‘˜=π‘‹π‘˜βˆ’1+β€–β€–π‘…π‘˜βˆ’1β€–β€–2β€–β€–π‘ˆπ‘˜βˆ’1β€–β€–2+β€–β€–π‘‰π‘˜βˆ’1β€–β€–2π‘ˆπ‘˜βˆ’1,π‘Œπ‘˜=π‘Œπ‘˜βˆ’1+β€–β€–π‘…π‘˜βˆ’1β€–β€–2β€–β€–π‘ˆπ‘˜βˆ’1β€–β€–2+β€–β€–π‘‰π‘˜βˆ’1β€–β€–2π‘‰π‘˜βˆ’1,π‘…π‘˜ξ€·ξ€·π‘‹=diagπ‘€βˆ’Ξ¦π‘˜,π‘Œπ‘˜ξ€Έξ€·π‘‹,π‘βˆ’Ξ¨π‘˜,π‘Œπ‘˜ξ€Έξ€Έ=π‘…π‘˜βˆ’1βˆ’β€–β€–π‘…π‘˜βˆ’1β€–β€–2β€–β€–π‘ˆπ‘˜βˆ’1β€–β€–2+β€–β€–π‘‰π‘˜βˆ’1β€–β€–2ξ€·Ξ¦ξ€·π‘ˆdiagπ‘˜βˆ’1,π‘‰π‘˜βˆ’1ξ€Έξ€·π‘ˆ,Ξ¨π‘˜βˆ’1,π‘‰π‘˜βˆ’1,π‘ˆξ€Έξ€Έπ‘˜=12ξ€Ίπ΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜,π‘Œπ‘˜π΅ξ€Έξ€Έπ‘‡+πΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜,π‘Œπ‘˜πΉξ€Έξ€Έπ‘‡+π‘ƒπ΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜,π‘Œπ‘˜π΅ξ€Έξ€Έπ‘‡π‘„+π‘ƒπΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜,π‘Œπ‘˜πΉξ€Έξ€Έπ‘‡π‘„ξ€»+β€–β€–π‘…π‘˜β€–β€–2β€–β€–π‘…π‘˜βˆ’1β€–β€–2π‘ˆπ‘˜βˆ’1,π‘‰π‘˜=12ξ€Ίβˆ’πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜βˆ’1,π‘Œπ‘˜βˆ’1π·ξ€Έξ€Έπ‘‡βˆ’πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜βˆ’1,π‘Œπ‘˜βˆ’1π»ξ€Έξ€Έπ‘‡βˆ’π‘…πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜βˆ’1,π‘Œπ‘˜βˆ’1π·ξ€Έξ€Έπ‘‡π‘†βˆ’π‘…πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜βˆ’1,π‘Œπ‘˜βˆ’1𝐻𝑇𝑆+β€–β€–π‘…π‘˜β€–β€–2β€–β€–π‘…π‘˜βˆ’1β€–β€–2π‘‰πΎβˆ’1.(2.3)

Step 5. Go to Step 3.
Obviously, it can be seen that π‘‹π‘˜,π‘ˆπ‘˜βˆˆπ‘…π‘Ÿπ‘šΓ—π‘›(𝑃,𝑄), π‘Œπ‘˜,π‘‰π‘˜βˆˆπ‘…π‘Ÿπ‘ Γ—π‘‘(𝑅,𝑆), where π‘˜=1,2,….

Lemma 2.2. For the sequences {𝑅𝑖}, {π‘ˆπ‘–}, and {𝑉𝑖} generated by Algorithm 2.1, and 𝑠β‰₯2, we have𝑅trπ‘‡π‘–π‘…π‘—ξ€Έξ€·π‘ˆ=0,trπ‘‡π‘–π‘ˆπ‘—+𝑉𝑇𝑖𝑉𝑗=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 [π‘‹βˆ—,π‘Œβˆ—] is an arbitrary solution pair of Problem 1, then for any initial generalized reflexive matrix pair [𝑋1,π‘Œ1], we have 𝑋trβˆ—βˆ’π‘‹π‘–ξ€Έπ‘‡π‘ˆπ‘–+ξ€·π‘Œβˆ—βˆ’π‘Œπ‘–ξ€Έπ‘‡π‘‰π‘–ξ‚=‖‖𝑅𝑖‖‖2,π‘˜=1,2,…,(2.5) where the sequences {𝑋𝑖},{π‘Œπ‘–}, {π‘ˆπ‘–}, {𝑉𝑖}, and {𝑅𝑖} 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 π‘˜ such that π‘ˆπ‘˜=0 and π‘‰π‘˜=0 but π‘…π‘˜β‰ 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,π‘Œ1]βˆˆβ„›π‘Ÿπ‘šΓ—π‘›(𝑃,𝑄)Γ—β„›π‘Ÿπ‘ Γ—π‘‘(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 π‘…π‘–β‰ πŸŽ,𝑖=1,2,…,π‘π‘ž+𝑠𝑑, by Lemma 2.3, we have π‘ˆπ‘–β‰ πŸŽ,π‘‰π‘–β‰ πŸŽ,𝑖=1,2,…,π‘π‘ž+𝑠𝑑, then we can compute [π‘‹π‘π‘ž+𝑠𝑑+1,π‘Œπ‘π‘ž+𝑠𝑑+1] by Algorithm 2.1.
By Lemma 2.2, we have𝑅trπ‘‡π‘π‘ž+𝑠𝑑+1𝑅𝑖𝑅=0,𝑖=1,2,…,π‘π‘ž+𝑠𝑑,tr𝑇𝑖𝑅𝑗=0,𝑖,𝑗=1,2,…,π‘π‘ž+𝑠𝑑,𝑖≠𝑗.(2.6)
It can be seen that the set of 𝑅1,𝑅2,…,π‘…π‘π‘ž+𝑠𝑑 is an orthogonal basis of the matrix subspace𝐿𝑆=𝐿∣𝐿=diag1,𝐿2ξ€Έ,𝐿1βˆˆβ„›π‘Γ—π‘ž,𝐿2βˆˆβ„›π‘ Γ—π‘‘ξ€Ύ,(2.7) which implies that π‘…π‘π‘ž+𝑠𝑑+1=0, that is, [π‘‹π‘π‘ž+𝑠𝑑+1,π‘Œπ‘π‘ž+𝑠𝑑+1]βˆˆβ„›π‘Ÿπ‘šΓ—π‘›(𝑃,𝑄)Γ—β„›π‘Ÿπ‘ Γ—π‘‘(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 𝐴π‘₯=𝑏 has a solution π‘₯βˆ—βˆˆπ‘…(𝐴𝑇), then π‘₯βˆ— 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=𝐴𝑇𝐾𝐡𝑇+𝐸𝑇𝐿𝐹𝑇+𝑃𝐴𝑇𝐾𝐡𝑇𝑄+𝑃𝐸𝑇𝐿𝐹𝑇𝑄 and π‘Œ1=βˆ’πΆπ‘‡πΎπ·π‘‡βˆ’πΊπ‘‡πΏπ»π‘‡βˆ’π‘…πΆπ‘‡πΎπ·π‘‡π‘† βˆ’π‘…πΊπ‘‡πΏπ»π‘‡π‘†, where πΎβˆˆβ„›π‘Γ—π‘ž,πΏβˆˆβ„›π‘˜Γ—π‘™ are arbitrary matrices, especially, 𝑋1=0βˆˆβ„›π‘šΓ—π‘›(𝑃,𝑄) and π‘Œ1=0βˆˆβ„›π‘ Γ—π‘‘(𝑅,𝑆), then the solution pair [π‘Œβˆ—,π‘βˆ—] 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: π΄π‘‹π΅βˆ’πΆπ‘Œπ·=𝑀,πΈπ‘‹πΉβˆ’πΊπ‘Œπ»=𝑁,π΄π‘ƒπ‘‹π‘„π΅βˆ’πΆπ‘…π‘Œπ‘†π·=𝑀,πΈπ‘ƒπ‘‹π‘„πΉβˆ’πΊπ‘…π‘Œπ‘†π»=𝑁.(2.8) Then, the system of matrix equations (2.8) is equivalent to βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽπ΅π‘‡βŠ—π΄βˆ’π·π‘‡πΉβŠ—πΆπ‘‡βŠ—πΈβˆ’π»π‘‡π΅βŠ—πΊπ‘‡π‘„βŠ—π΄π‘ƒβˆ’π·π‘‡πΉπ‘†βŠ—πΆπ‘…π‘‡π‘„βŠ—πΈπ‘ƒβˆ’π»π‘‡βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ =βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ π‘†βŠ—πΊπ‘…vec(𝑋)vec(π‘Œ)vec(𝑀)vec(𝑁)vec(𝑀)vec(𝑁).(2.9) Let 𝑋1=𝐴𝑇𝐾𝐡𝑇+𝐸𝑇𝐿𝐹𝑇+𝑃𝐴𝑇𝐾𝐡𝑇𝑄+𝑃𝐸𝑇𝐿𝐹𝑇𝑄 and π‘Œ1=βˆ’πΆπ‘‡πΎπ·π‘‡βˆ’πΊπ‘‡πΏπ»π‘‡ βˆ’ π‘…πΆπ‘‡πΎπ·π‘‡π‘†βˆ’π‘…πΊπ‘‡πΏπ»π‘‡π‘†, where πΎβˆˆβ„›π‘Γ—π‘ž,πΏβˆˆβ„›π‘˜Γ—π‘™ are arbitrary matrices, then βŽ›βŽœβŽœβŽξ€·π‘‹vec1ξ€Έξ€·π‘Œvec1ξ€ΈβŽžβŽŸβŽŸβŽ =βŽ›βŽœβŽœβŽξ€·π΄vec𝑇𝐾𝐡𝑇+𝐸𝑇𝐿𝐹𝑇+𝑃𝐴𝑇𝐾𝐡𝑇𝑄+𝑃𝐸𝑇𝐿𝐹𝑇𝑄vecβˆ’πΆπ‘‡πΎπ·π‘‡βˆ’πΊπ‘‡πΏπ»π‘‡βˆ’π‘…πΆπ‘‡πΎπ·π‘‡π‘†βˆ’π‘…πΊπ‘‡πΏπ»π‘‡π‘†ξ€ΈβŽžβŽŸβŽŸβŽ =βŽ›βŽœβŽœβŽπ΅βŠ—π΄π‘‡πΉβŠ—πΈπ‘‡π‘„π΅βŠ—π‘ƒπ΄π‘‡π‘„πΉβŠ—π‘ƒπΈπ‘‡βˆ’π·βŠ—πΆπ‘‡βˆ’π»βŠ—πΊπ‘‡βˆ’π‘†π·βŠ—π‘…πΆπ‘‡βˆ’π‘†π»βŠ—π‘…πΊπ‘‡βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ =βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽπ΅vec(𝐾)vec(𝐿)vec(𝐾)vec(𝐿)π‘‡βŠ—π΄βˆ’π·π‘‡πΉβŠ—πΆπ‘‡βŠ—πΈβˆ’π»π‘‡π΅βŠ—πΊπ‘‡π‘„βŠ—π΄π‘ƒβˆ’π·π‘‡πΉπ‘†βŠ—πΆπ‘…π‘‡π‘„βŠ—πΈπ‘ƒβˆ’π»π‘‡βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ π‘†βŠ—πΊπ‘…π‘‡βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽπ΅vec(𝐾)vec(𝐺)vec(𝐾)vec(𝐺)βˆˆβ„›π‘‡βŠ—π΄βˆ’π·π‘‡πΉβŠ—πΆπ‘‡βŠ—πΈβˆ’π»π‘‡π΅βŠ—πΊπ‘‡π‘„βŠ—π΄π‘ƒβˆ’π·π‘‡πΉπ‘†βŠ—πΆπ‘…π‘‡π‘„βŠ—πΈπ‘ƒβˆ’π»π‘‡βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ π‘†βŠ—πΊπ‘…π‘‡βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ .(2.10)Furthermore, we can see that all π‘‹π‘˜,π‘Œπ‘˜ generated by Algorithm 2.1 satisfy βŽ›βŽœβŽœβŽξ€·π‘‹vecπ‘˜ξ€Έξ€·π‘Œvecπ‘˜ξ€ΈβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽπ΅βˆˆβ„›π‘‡βŠ—π΄βˆ’π·π‘‡πΉβŠ—πΆπ‘‡βŠ—πΈβˆ’π»π‘‡π΅βŠ—πΊπ‘‡π‘„βŠ—π΄π‘ƒβˆ’π·π‘‡πΉπ‘†βŠ—πΆπ‘…π‘‡π‘„βŠ—πΈπ‘ƒβˆ’π»π‘‡βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ π‘†βŠ—πΊπ‘…π‘‡βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,(2.11)by Lemma 2.6, we know that [π‘‹βˆ—,π‘Œβˆ—] is the least Frobenius norm generalized reflexive solution pair of the system of linear equations (2.9). Since vector operator is isomorphic, [π‘‹βˆ—,π‘Œβˆ—] is the unique least Frobenius norm generalized reflexive solution pair of the system of matrix equations (2.8), then [π‘‹βˆ—,π‘Œβˆ—] 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 𝑆𝐸 is not empty. For a given matrix pair [𝑋0,π‘Œ0]βˆˆβ„›π‘Ÿπ‘šΓ—π‘›(𝑃,𝑄)Γ—β„›π‘Ÿπ‘ Γ—π‘‘(𝑅,𝑆), we haveξ‚»βŸΊξ‚»π΄ξ€·π΄π‘‹π΅βˆ’πΆπ‘Œπ·=π‘€πΈπ‘‹πΉβˆ’πΊπ‘Œπ»=π‘π‘‹βˆ’π‘‹0ξ€Έξ€·π΅βˆ’πΆπ‘Œβˆ’π‘Œ0𝐷=π‘€βˆ’π΄π‘‹0𝐡+πΆπ‘Œ0π·πΈξ€·π‘‹βˆ’π‘‹0ξ€Έξ€·πΉβˆ’πΊπ‘Œβˆ’π‘Œ0𝐻=π‘βˆ’πΈπ‘‹0𝐹+πΊπ‘Œ0𝐻(3.1) Set 𝑋=π‘‹βˆ’π‘‹0,ξ‚π‘Œ=π‘Œβˆ’π‘Œ0,𝑀=π‘€βˆ’π΄π‘‹0𝐡+πΆπ‘Œ0𝐷,𝑁=π‘βˆ’πΈπ‘‹0𝐹+πΊπ‘Œ0𝐻, then Problem 2 is equivalent to that of finding the least Frobenius norm generalized reflexive solutions pair [ξ‚π‘‹βˆ—,ξ‚π‘Œβˆ—] of the corresponding generalized coupled Sylvester matrix equationsπ΄ξ‚ξ‚ξ‚‹πΈξ‚ξ‚ξ‚π‘‹π΅βˆ’πΆπ‘Œπ·=𝑀,π‘‹πΉβˆ’πΊπ‘Œπ»=𝑁.(3.2) By using Algorithm 2.1, let initial iteration matrix 𝑋1=𝐴𝑇𝐾𝐡𝑇+𝐸𝑇𝐿𝐹𝑇+𝑃𝐴𝑇𝐾𝐡𝑇𝑄+𝑃𝐸𝑇𝐿𝐹𝑇𝑄 and ξ‚π‘Œ1=βˆ’πΆπ‘‡πΎπ·π‘‡βˆ’πΊπ‘‡πΏπ»π‘‡βˆ’π‘…πΆπ‘‡πΎπ·π‘‡π‘†βˆ’π‘…πΊπ‘‡πΏπ»π‘‡π‘†, or more especially, let 𝑋1=0βˆˆβ„›π‘Ÿπ‘šΓ—π‘›(𝑃,𝑄) and ξ‚π‘Œ1=0βˆˆβ„›π‘Ÿπ‘ Γ—π‘‘(𝑅,𝑆), then we can get the least Frobenius norm generalized reflexive solution pair [ξ‚π‘‹βˆ—,ξ‚π‘Œβˆ—] of (3.2). Thus, the generalized reflexive solution pair of the problem 2 can be represented as [𝑋𝑋,π‘Œ]=[βˆ—+𝑋0,ξ‚π‘Œβˆ—+π‘Œ0].

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 π΄π‘‹π΅βˆ’πΆπ‘Œπ·=𝑀,πΈπ‘‹π‘Œβˆ’πΊπ‘Œπ»=𝑁, where βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ .𝐴=13βˆ’57βˆ’92046βˆ’10βˆ’296βˆ’83622βˆ’3βˆ’55βˆ’22βˆ’1βˆ’1184βˆ’6βˆ’9βˆ’9,𝐡=48βˆ’54βˆ’15βˆ’23392βˆ’6βˆ’27βˆ’81𝐢=6βˆ’57βˆ’9246βˆ’119βˆ’123βˆ’81364βˆ’15βˆ’515βˆ’13βˆ’1129βˆ’6βˆ’9,𝐷=718βˆ’6βˆ’45βˆ’233βˆ’12081694βˆ’58βˆ’29𝐸=145βˆ’171βˆ’23βˆ’2541342βˆ’36βˆ’81βˆ’548,𝐹=13βˆ’582βˆ’115βˆ’6251327βˆ’97βˆ’96βˆ’5121𝐺=12βˆ’58βˆ’55βˆ’73249βˆ’6βˆ’37βˆ’1211,𝐻=248βˆ’547βˆ’15βˆ’236392βˆ’65βˆ’27βˆ’8114βˆ’3βˆ’26𝑀=519117717011632βˆ’1031583βˆ’100238282180010293308βˆ’514839βˆ’4932458βˆ’75311322683βˆ’762βˆ’1164258858408,𝑁=βˆ’2426964βˆ’26532092603βˆ’65247βˆ’919291788βˆ’13311547βˆ’17992712βˆ’1684βˆ’659βˆ’27301756βˆ’765(4.1)Let βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ π‘ƒ=0000βˆ’10001000βˆ’10001000βˆ’10000,𝑄=0010000βˆ’110000βˆ’100𝑅=0001010000βˆ’101000,𝑆=000100000100βˆ’1001000001000(4.2)be generalized reflection matrices.
We will find the generalized reflexive solutions of the matrix equations π΄π‘‹π΅βˆ’πΆπ‘Œπ·=𝑀,πΈπ‘‹π‘Œβˆ’πΊπ‘Œπ»=𝑁 by using Algorithm 2.1. It can be verified that the matrix equations are consistent over generalized reflexive matrices and the solutions areπ‘‹βˆ—=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βˆ’29253111βˆ’173βˆ’731113βˆ’1βˆ’2529,π‘Œβˆ—=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ 1416βˆ’13497097βˆ’3βˆ’8βˆ’8383411416.(4.3) Because of the influence of the error of calculation, the residual 𝑅𝑖 is usually unequal to zero in the process of the iteration, where 𝑖=1,2,…. For any chosen positive number πœ€; however, small enough, for example, πœ€=1.0000π‘’βˆ’010, whenever β€–π‘…π‘˜β€–<πœ€, stop the iteration, π‘‹π‘˜ and π‘Œπ‘˜ are regarded to be generalized reflexive solutions of the matrix equations π΄π‘‹π΅βˆ’πΆπ‘Œπ·=𝑀,πΈπ‘‹π‘Œβˆ’πΊπ‘Œπ»=𝑁. Choose an initially iterative matrix pair [𝑋1,π‘Œ1]βˆˆβ„›π‘Ÿ5Γ—4(𝑃,𝑄)Γ—β„›π‘Ÿ4Γ—5(𝑅,𝑆), such as 𝑋1=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βˆ’12246βˆ’13278βˆ’783βˆ’261βˆ’2412,π‘Œ1=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ 34βˆ’13791091βˆ’31βˆ’83βˆ’137134.(4.4) By Algorithm 2.1, we have 𝑋30=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,π‘Œβˆ’2.00009.00002.00005.00003.00001.000011.0000βˆ’1.00007.00003.0000βˆ’7.00003.000011.00001.00003.0000βˆ’1.0000βˆ’2.00005.00002.00009.000030=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,‖‖𝑅14.000016.0000βˆ’1.00003.00004.00009.00007.000009.00007.0000βˆ’3.0000βˆ’8.0000βˆ’8.00003.00008.00003.00004.00001.000014.000016.000030β€–β€–=2.9703π‘’βˆ’012<πœ€.(4.5)So we obtain the generalized reflexive solutions of the matrix equations π΄π‘‹π΅βˆ’πΆπ‘Œπ·=𝑀,πΈπ‘‹π‘Œβˆ’πΊπ‘Œπ»=𝑁. The relative error of the solutions and the residual are shown in Figure 1, where the relative error π‘…πΈπ‘˜=(β€–π‘‹π‘˜βˆ’π‘‹βˆ—β€–+β€–π‘Œπ‘˜βˆ’π‘Œβˆ—β€–)/(β€–π‘‹βˆ—β€–+β€–π‘Œβˆ—β€–) and the residual π‘…π‘˜=β€–π‘…π‘˜β€–.
Let𝑋1=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ 00000000000000000000,π‘Œ1=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ 00000000000000000000,(4.6) by Algorithm 2.1, we have π‘‹βˆ—=𝑋30=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,π‘Œβˆ’2.00009.00002.00005.00003.00001.000011.0000βˆ’1.00007.00003.0000βˆ’7.00003.000011.00001.00003.0000βˆ’1.0000βˆ’2.00005.00002.00009.0000βˆ—=π‘Œ30=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,‖‖𝑅14.000016.0000βˆ’1.00003.00004.00009.00007.000009.00007.0000βˆ’3.0000βˆ’8.0000βˆ’8.00003.00008.00003.00004.00001.000014.000016.000030β€–β€–=8.2565π‘’βˆ’012<πœ€.(4.7)The relative error of the solutions and the residual are shown in Figure 2.

152805.fig.001
Figure 1: The relative error of the solutions and the residual for Example 4.1 with 𝑋1β‰ 0,π‘Œ1β‰ 0.
152805.fig.002
Figure 2: The relative error of the solutions and the residual for Example 4.1 with 𝑋1=0,π‘Œ1=0.

Example 4.2. Consider the unique least-norm generalized reflexive solutions of the matrix equations in Example 4.1. Let βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,𝑋𝐾=10120βˆ’1011βˆ’101201βˆ’30121βˆ’10βˆ’2βˆ’1,𝐿=βˆ’11βˆ’10501βˆ’1321βˆ’1βˆ’203201βˆ’361=𝐴𝑇𝐾𝐡𝑇+𝐢𝑇𝐿𝐷𝑇+𝑃𝐴𝑇𝐾𝐡𝑇𝑄+π‘ƒπΆπ‘‡πΏπ·π‘‡π‘Œπ‘„,1=βˆ’πΈπ‘‡πΎπΉπ‘‡βˆ’πΊπ‘‡πΏπ»π‘‡βˆ’π‘…πΈπ‘‡πΎπΉπ‘‡π‘†βˆ’π‘…πΊπ‘‡πΏπ»π‘‡π‘†.(4.8)By using Algorithm 2.1, we have the least-norm generalized reflexive solutions of the matrix equations π΄π‘‹π΅βˆ’πΆπ‘Œπ·=𝑀,πΈπ‘‹π‘Œβˆ’πΊπ‘Œπ»=𝑁 as follows: π‘‹βˆ—=𝑋30=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,π‘Œβˆ’2.00009.00002.00005.00003.00001.000011.0000βˆ’1.00007.00003.0000βˆ’7.00003.000011.00001.00003.0000βˆ’1.0000βˆ’2.00005.00002.00009.0000βˆ—=π‘Œ30=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,‖‖𝑅14.000016.0000βˆ’1.00003.00004.00009.00007.000009.00007.0000βˆ’3.0000βˆ’8.0000βˆ’8.00003.00008.00003.00004.00001.000014.000016.000030β€–β€–=2.3986π‘’βˆ’011π‘’βˆ’012<πœ€.(4.9)
The relative error of the solutions and the residual are shown in Figure 3.

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Figure 3: The relative error of the solutions and the residual for Example 4.2.

Example 4.3. Let 𝑆𝐸 denote the set of all generalized reflexive solutions of the matrix equations in Example 4.1. For a given generalized reflexive matrices 𝑋0=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ 3βˆ’1223βˆ’2001βˆ’3βˆ’1βˆ’30032βˆ’22βˆ’3βˆ’1,π‘Œ0=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ 24βˆ’220130135βˆ’22βˆ’5220224,(4.10)we will find [𝑋,π‘Œ]βˆˆπ‘†πΈ, such that β€–β€–ξπ‘‹βˆ’π‘‹0β€–β€–+β€–β€–ξπ‘Œβˆ’π‘Œ0β€–β€–=min[]𝑋,π‘Œβˆˆπ‘†πΈβ€–β€–π‘‹βˆ’π‘‹0β€–β€–+β€–β€–π‘Œβˆ’π‘Œ0β€–β€–,(4.11)that is, find the optimal approximate generalized reflexive solution pair to the matrix pair [𝑋0,π‘Œ0] in 𝑆𝐸 in Frobenius norm.
Let 𝑋=π‘‹βˆ’π‘‹0,ξ‚π‘Œ=π‘Œβˆ’π‘Œ0,𝑀=π‘€βˆ’π΄π‘‹0𝐡+πΆπ‘Œ0𝐷,𝑁=π‘βˆ’πΈπ‘‹0𝐹+πΊπ‘Œ0𝐻, by the method mentioned in Section 3, we can obtain the least-norm generalized reflexive solution pair [ξ‚π‘‹βˆ—,ξ‚π‘Œβˆ—] of the matrix equations 𝐴𝑁𝑋𝐡+πΆπ‘Œπ·=𝑀,𝐸𝑋𝐹+πΊπ‘Œπ»= by choosing the initial iteration matrices 𝑋1=0 and ξ‚π‘Œ1=0, then by Algorithm 2.1, we have thatξ‚π‘‹βˆ—=ξ‚π‘‹βˆ—30=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,ξ‚π‘Œβˆ’5.000010.00000.00003.0000βˆ’0.00003.000011.0000βˆ’1.00006.00006.0000βˆ’6.00006.000011.00001.0000βˆ’0.0000βˆ’3.0000βˆ’0.00003.00005.000010.0000βˆ—=ξ‚π‘Œβˆ—30=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,‖‖𝑅12.000012.00001.00001.00004.00008.00004.000008.00004.0000βˆ’8.0000βˆ’6.0000βˆ’10.00008.00006.00001.00004.0000βˆ’1.000012.000012.000030β€–β€–=6.3482π‘’βˆ’010<πœ€=1.0000π‘’βˆ’010(4.12)and the optimal approximate generalized reflexive solutions to the matrix pair [𝑋0,π‘Œ0] in Frobenius norm are 𝑋𝑋=βˆ—30+𝑋0=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,ξξ‚π‘Œβˆ’2.00009.00002.00005.00003.00001.000011.0000βˆ’1.00007.00003.0000βˆ’7.00003.000011.00001.00003.0000βˆ’1.0000βˆ’2.00005.00002.00009.0000π‘Œ=βˆ—30+π‘Œ0=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ .14.000016.0000βˆ’1.00003.00004.00009.00007.000009.00007.0000βˆ’3.0000βˆ’8.0000βˆ’8.00003.00008.00003.00004.00001.000014.000016.0000(4.13)The relative error of the solutions and the residual are shown in Figure 4, where the relative error ξ‚π‘‹π‘…πΈπ‘˜=(β€–π‘˜+𝑋0βˆ’π‘‹βˆ—ξ‚π‘Œβ€–+β€–π‘˜+π‘Œ0βˆ’π‘Œβˆ—β€–)/(β€–π‘‹βˆ—β€–+β€–π‘Œβˆ—β€–) and the residual π‘…π‘˜=β€–π‘…π‘˜β€–.

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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 π΄π‘‹π΅βˆ’πΆπ‘Œπ·=𝑀,πΈπ‘‹π‘Œβˆ’πΊπ‘Œπ»=𝑁 over generalized reflexive matrix pair [𝑋,π‘Œ]βˆˆβ„›π‘Ÿπ‘šΓ—π‘›(𝑃,𝑄)Γ—β„›π‘Ÿπ‘ Γ—π‘‘(𝑅,𝑆). When the matrix equations π΄π‘‹π΅βˆ’πΆπ‘Œπ·=𝑀,πΈπ‘‹π‘Œβˆ’πΊπ‘Œπ»=𝑁 are consistent over generalized reflexive matrices 𝑋 and π‘Œ, for any generalized reflexive initial iterative matrix pair [𝑋1,π‘Œ1]βˆˆβ„›π‘Ÿπ‘šΓ—π‘›(𝑃,𝑄)Γ—β„›π‘Ÿπ‘ Γ—π‘‘(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=𝐴𝑇𝐾𝐡𝑇+𝐸𝑇𝐿𝐹𝑇+𝑃𝐴𝑇𝐾𝐡𝑇𝑄+𝑃𝐸𝑇𝐿𝐹𝑇𝑄 and π‘Œ1=βˆ’πΆπ‘‡πΎπ·π‘‡βˆ’πΊπ‘‡πΏπ»π‘‡βˆ’π‘…πΆπ‘‡πΎπ·π‘‡π‘†βˆ’π‘…πΊπ‘‡πΏπ»π‘‡π‘†, where πΎβˆˆβ„›π‘Γ—π‘ž,πΏβˆˆβ„›π‘˜Γ—π‘™ are arbitrary matrices, especially, let 𝑋1=0βˆˆβ„›π‘Ÿπ‘šΓ—π‘›(𝑃,𝑄) and π‘Œ1=0βˆˆβ„›π‘Ÿπ‘ Γ—π‘‘(𝑅,𝑆), the unique least-norm generalized reflexive solutions of the matrix equations can be derived. Furthermore, the optimal approximate solutions of π΄π‘‹π΅βˆ’πΆπ‘Œπ·=𝑀,πΈπ‘‹π‘Œβˆ’πΊπ‘Œπ»=𝑁 for a given generalized reflexive matrix pair [𝑋0,π‘Œ0]βˆˆβ„›π‘Ÿπ‘šΓ—π‘›(𝑃,𝑄)Γ—β„›π‘Ÿπ‘ Γ—π‘‘(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 𝐡=𝐼,𝐢=𝐼,𝐹=𝐼,𝐺=𝐼, 𝑃=𝑄, and 𝑅=𝑆, then our results reduce to those in [7]. When 𝑃=𝑄,𝑅=𝑆,𝑋𝑇=𝑋, and π‘Œπ‘‡=π‘Œ, the results in this paper reduce to those in [12]. When 𝐡=𝐼,𝐢=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(𝑅𝑇𝑖𝑅𝑗)=tr(𝑅𝑇𝑗𝑅𝑖), tr(π‘ˆπ‘‡π‘–π‘ˆπ‘—)=tr(π‘ˆπ‘‡π‘—π‘ˆπ‘–), and tr(𝑉𝑇𝑖𝑉𝑗)=tr(𝑉𝑇𝑗𝑉𝑖) for all 𝑖,𝑗=1,2,…,𝑠, we only need to prove that𝑅trπ‘‡π‘–π‘…π‘—ξ€Έξ€·π‘ˆ=0,trπ‘‡π‘–π‘ˆπ‘—+𝑉𝑇𝑖𝑉𝑗=0,1≀𝑗<𝑖≀𝑠.(A.1) We prove the conclusion by induction, and two steps are required.

Step 1. We will show that 𝑅tr𝑇𝑖+1π‘…π‘–ξ€Έξ€·π‘ˆ=0,tr𝑇𝑖+1π‘ˆπ‘–+𝑉𝑇𝑖+1𝑉𝑖=0,𝑖=1,2,…,π‘ βˆ’1.(A.2)
To prove this conclusion, we also use induction.
For 𝑖=1, by Algorithm 2.1, we have that 𝑅tr𝑇2𝑅1ξ€ΈβŽ›βŽœβŽœβŽξƒ¬π‘…=tr1βˆ’β€–β€–π‘…1β€–β€–2β€–β€–π‘ˆ1β€–β€–2+‖‖𝑉1β€–β€–2ξ€·Ξ¦ξ€·π‘ˆdiag1,𝑉1ξ€Έξ€·π‘ˆ,Ξ¨1,𝑉1𝑇𝑅1⎞⎟⎟⎠=‖‖𝑅1β€–β€–2βˆ’β€–β€–π‘…1β€–β€–2β€–β€–π‘ˆ1β€–β€–2+‖‖𝑉1β€–β€–2ξ‚€ξ€·ξ€·Ξ¦ξ€·π‘ˆtrdiag1,𝑉1ξ€Έξ€·π‘ˆ,Ξ¨1,𝑉1𝑇𝑋×diagπ‘€βˆ’Ξ¦1,π‘Œ1𝑋,π‘βˆ’Ξ¨1,π‘Œ1=‖‖𝑅1β€–β€–2βˆ’β€–β€–π‘…1β€–β€–2β€–β€–π‘ˆ1β€–β€–2+‖‖𝑉1β€–β€–2ξ‚€ξ€·Ξ¦ξ€·π‘ˆΓ—tr1,𝑉1ξ€Έξ€Έπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦1,π‘Œ1+ξ€·Ξ¨ξ€·π‘ˆξ€Έξ€Έ1,𝑉1ξ€Έξ€Έπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨1,π‘Œ1=‖‖𝑅1β€–β€–2βˆ’β€–β€–π‘…1β€–β€–2β€–β€–π‘ˆ1β€–β€–2+‖‖𝑉1β€–β€–2ξ€·π‘ˆtr𝑇1π΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦1,π‘Œ1𝐡𝑇+π‘ˆπ‘‡1πΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨1,π‘Œ1πΉξ€Έξ€Έπ‘‡βˆ’π‘‰π‘‡1πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦1,π‘Œ1π·ξ€Έξ€Έπ‘‡βˆ’π‘‰π‘‡1πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨1,π‘Œ1𝐻𝑇=‖‖𝑅1β€–β€–2βˆ’β€–β€–π‘…1β€–β€–2β€–β€–π‘ˆ1β€–β€–2+‖‖𝑉1β€–β€–2ξƒ©π‘ˆΓ—tr𝑇1ξƒ¬π΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦1,π‘Œ1𝐡𝑇+πΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨1,π‘Œ1𝐹𝑇2+π΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦1,π‘Œ1𝐡𝑇+πΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨1,π‘Œ1𝐹𝑇2+π‘ƒπ΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦1,π‘Œ1𝐡𝑇𝑄+π‘ƒπΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨1,π‘Œ1𝐹𝑇𝑄2βˆ’π‘ƒπ΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦1,π‘Œ1𝐡𝑇𝑄+π‘ƒπΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨1,π‘Œ1𝐹𝑇𝑄2ξƒ­+𝑉𝑇1ξƒ¬βˆ’πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦1,π‘Œ1π·ξ€Έξ€Έπ‘‡βˆ’πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨1,π‘Œ1𝐻𝑇2+βˆ’πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦1,π‘Œ1π·ξ€Έξ€Έπ‘‡βˆ’πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨1,π‘Œ1𝐻𝑇2+βˆ’π‘…πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦1,π‘Œ1π·ξ€Έξ€Έπ‘‡π‘†βˆ’π‘…πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨1,π‘Œ1𝐻𝑇𝑆2βˆ’βˆ’π‘…πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦1,π‘Œ1π·ξ€Έξ€Έπ‘‡π‘†βˆ’π‘…πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨1,π‘Œ1𝐻𝑇𝑆2=‖‖𝑅ξƒͺ1β€–β€–2βˆ’β€–β€–π‘…1β€–β€–2β€–β€–π‘ˆ1β€–β€–2+‖‖𝑉1β€–β€–2ξƒ©π‘ˆΓ—tr𝑇1ξƒ¬π΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦1,π‘Œ1𝐡𝑇+πΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨1,π‘Œ1𝐹𝑇2+π‘ƒπ΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦1,π‘Œ1𝐡𝑇𝑄+π‘ƒπΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨1,π‘Œ1𝐹𝑇𝑄2ξƒ­+𝑉𝑇1ξƒ¬βˆ’πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦1,π‘Œ1π·ξ€Έξ€Έπ‘‡βˆ’πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨1,π‘Œ1𝐻𝑇2+βˆ’π‘…πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦1,π‘Œ1π·ξ€Έξ€Έπ‘‡π‘†βˆ’π‘…πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨1,π‘Œ1𝐻𝑇𝑆2=‖‖𝑅ξƒͺ1β€–β€–2βˆ’β€–β€–π‘…1β€–β€–2β€–β€–π‘ˆ1β€–β€–2+‖‖𝑉1β€–β€–2ξ€·π‘ˆtr𝑇1π‘ˆ1+𝑉𝑇1𝑉1ξ€Έξ€·π‘ˆ=0,tr𝑇2π‘ˆ1𝑉+tr𝑇2𝑉1𝐴=trξƒ©ξƒ¬π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦2,π‘Œ2𝐡𝑇+πΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨2,π‘Œ2𝐹𝑇2+π‘ƒπ΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦2,π‘Œ2𝐡𝑇𝑄+π‘ƒπΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨2,π‘Œ2𝐹𝑇𝑄2+‖‖𝑅2β€–β€–2‖‖𝑅1β€–β€–2π‘ˆ1ξƒ­π‘‡π‘ˆ1⎞⎟⎟⎠+trξƒ©ξƒ¬βˆ’πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦2,π‘Œ2π·ξ€Έξ€Έπ‘‡βˆ’πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨2,π‘Œ2𝐻𝑇2+βˆ’π‘…πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦2,π‘Œ2π·ξ€Έξ€Έπ‘‡π‘†βˆ’π‘…πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨2,π‘Œ2𝐻𝑇𝑆2+‖‖𝑅2β€–β€–2‖‖𝑅1β€–β€–2𝑉1𝑇𝑉1βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽξƒ¬π΄=trπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦2,π‘Œ2𝐡𝑇+πΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨2,π‘Œ2𝐹𝑇+‖‖𝑅2β€–β€–2‖‖𝑅1β€–β€–2π‘ˆ1ξƒ­π‘‡π‘ˆ1βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽξƒ¬+trβˆ’πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦2,π‘Œ2π·ξ€Έξ€Έπ‘‡βˆ’πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨2,π‘Œ2𝐻𝑇+‖‖𝑅2β€–β€–2‖‖𝑅1β€–β€–2𝑉1𝑇𝑉1βŽžβŽŸβŽŸβŽ ξ€·π‘ˆ=tr𝑇1ξ€Ίπ΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦2,π‘Œ2𝐡𝑇+πΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨2,π‘Œ2𝐹𝑇+𝑉𝑇1ξ€Ίβˆ’πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦2,π‘Œ2π·ξ€Έξ€Έπ‘‡βˆ’πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨2,π‘Œ2𝐻𝑇+‖‖𝑅2β€–β€–2‖‖𝑅1β€–β€–2ξ‚€β€–β€–π‘ˆ1β€–β€–2+‖‖𝑉1β€–β€–2𝑋=trπ‘€βˆ’Ξ¦2,π‘Œ2ξ€Έξ€Έπ‘‡π΄π‘ˆ1𝑋𝐡+π‘βˆ’Ξ¨2,π‘Œ2ξ€Έξ€Έπ‘‡πΈπ‘ˆ1ξ€·ξ€·π‘‹πΉβˆ’π‘€βˆ’Ξ¦2,π‘Œ2𝑇𝐢𝑉1π·βˆ’ξ€·ξ€·π‘‹π‘βˆ’Ξ¨2,π‘Œ2𝑇𝐺𝑉1𝐻+‖‖𝑅2β€–β€–2‖‖𝑅1β€–β€–2ξ‚€β€–β€–π‘ˆ1β€–β€–2+‖‖𝑉1β€–β€–2𝑋=trdiagπ‘€βˆ’Ξ¦2,π‘Œ2𝑇,ξ€·ξ€·π‘‹π‘βˆ’Ξ¨2,π‘Œ2ξ€Έξ€Έπ‘‡ξ‚ξ€·Ξ¦ξ€·π‘ˆdiag1,𝑉1ξ€Έξ€·π‘ˆ,Ξ¨1,𝑉1+‖‖𝑅2β€–β€–2‖‖𝑅1β€–β€–2ξ‚€β€–β€–π‘ˆ1β€–β€–2+‖‖𝑉1β€–β€–2=β€–β€–π‘ˆ1β€–β€–2+‖‖𝑉1β€–β€–2‖‖𝑅1β€–β€–2𝑅tr𝑇2𝑅1βˆ’π‘…2+‖‖𝑅2β€–β€–2‖‖𝑅1β€–β€–2ξ‚€β€–β€–π‘ˆ1β€–β€–2+‖‖𝑉1β€–β€–2=0.(A.3)
Assume that (A.2) holds for 𝑖=π‘˜βˆ’1, that is, tr(π‘…π‘‡π‘˜π‘…π‘˜βˆ’1)=0,tr(π‘ˆπ‘‡π‘˜π‘ˆπ‘˜βˆ’1+π‘‰π‘‡π‘˜π‘‰π‘˜βˆ’1)=0.
When 𝑖=π‘˜, we have that 𝑅trπ‘‡π‘˜+1π‘…π‘˜ξ€ΈβŽ›βŽœβŽœβŽξƒ¬π‘…=trπ‘˜βˆ’β€–β€–π‘…π‘˜β€–β€–2β€–β€–π‘ˆπ‘˜β€–β€–2+β€–β€–π‘‰π‘˜β€–β€–2ξ€·Ξ¦ξ€·π‘ˆdiagπ‘˜,π‘‰π‘˜ξ€Έξ€·π‘ˆ,Ξ¨π‘˜,π‘‰π‘˜ξƒ­ξ€Έξ€Έπ‘‡π‘…π‘˜βŽžβŽŸβŽŸβŽ =β€–β€–π‘…π‘˜β€–β€–2βˆ’β€–β€–π‘…π‘˜β€–β€–2β€–β€–π‘ˆπ‘˜β€–β€–2+β€–β€–π‘‰π‘˜β€–β€–2ξ€·ξ€·Ξ¦ξ€·π‘ˆtrdiagπ‘˜,π‘‰π‘˜ξ€Έξ€·π‘ˆ,Ξ¨π‘˜,π‘‰π‘˜ξ€Έξ€Έξ€Έπ‘‡ξ€·ξ€·π‘‹Γ—diagπ‘€βˆ’Ξ¦π‘˜,π‘Œπ‘˜ξ€Έξ€·π‘‹,π‘βˆ’Ξ¨π‘˜,π‘Œπ‘˜=β€–β€–π‘…ξ€Έξ€Έπ‘˜β€–β€–2βˆ’β€–β€–π‘…π‘˜β€–β€–2β€–β€–π‘ˆπ‘˜β€–β€–2+β€–β€–π‘‰π‘˜β€–β€–2ξ‚€ξ€·Ξ¦ξ€·π‘ˆΓ—trπ‘˜,π‘‰π‘˜ξ€Έξ€Έπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜,π‘Œπ‘˜+ξ€·Ξ¨ξ€·π‘ˆξ€Έξ€Έπ‘˜,π‘‰π‘˜ξ€Έξ€Έπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜,π‘Œπ‘˜ξ‚=β€–β€–π‘…ξ€Έξ€Έπ‘˜β€–β€–2βˆ’β€–β€–π‘…π‘˜β€–β€–2β€–β€–π‘ˆπ‘˜β€–β€–2+β€–β€–π‘‰π‘˜β€–β€–2ξ€·π‘ˆtrπ‘‡π‘˜π΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜,π‘Œπ‘˜π΅ξ€Έξ€Έπ‘‡+π‘ˆπ‘‡π‘˜πΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜,π‘Œπ‘˜πΉξ€Έξ€Έπ‘‡βˆ’π‘‰π‘‡π‘˜πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜,π‘Œπ‘˜π·ξ€Έξ€Έπ‘‡βˆ’π‘‰π‘‡π‘˜πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜,π‘Œπ‘˜π»ξ€Έξ€Έπ‘‡ξ€Έ=β€–β€–π‘…π‘˜β€–β€–2βˆ’β€–β€–π‘…π‘˜β€–β€–2β€–β€–π‘ˆπ‘˜β€–β€–2+β€–β€–π‘‰π‘˜β€–β€–2ξƒ©π‘ˆtrπ‘‡π‘˜ξƒ¬π΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜,π‘Œπ‘˜π΅ξ€Έξ€Έπ‘‡+πΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜,π‘Œπ‘˜πΉξ€Έξ€Έπ‘‡2+π΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜,π‘Œπ‘˜π΅ξ€Έξ€Έπ‘‡+πΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜,π‘Œπ‘˜πΉξ€Έξ€Έπ‘‡2+π‘ƒπ΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜,π‘Œπ‘˜π΅ξ€Έξ€Έπ‘‡π‘„+π‘ƒπΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜,π‘Œπ‘˜πΉξ€Έξ€Έπ‘‡π‘„2βˆ’π‘ƒπ΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜,π‘Œπ‘˜π΅ξ€Έξ€Έπ‘‡π‘„+π‘ƒπΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜,π‘Œπ‘˜πΉξ€Έξ€Έπ‘‡π‘„2ξƒ­+π‘‰π‘‡π‘˜ξƒ¬βˆ’πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜,π‘Œπ‘˜π·ξ€Έξ€Έπ‘‡βˆ’πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜,π‘Œπ‘˜π»ξ€Έξ€Έπ‘‡2+βˆ’πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜,π‘Œπ‘˜π·ξ€Έξ€Έπ‘‡βˆ’πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜,π‘Œπ‘˜π»ξ€Έξ€Έπ‘‡2+βˆ’π‘…πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜,π‘Œπ‘˜π·ξ€Έξ€Έπ‘‡π‘†βˆ’π‘…πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜,π‘Œπ‘˜π»ξ€Έξ€Έπ‘‡π‘†2βˆ’βˆ’π‘…πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜,π‘Œπ‘˜π·ξ€Έξ€Έπ‘‡π‘†βˆ’π‘…πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜,π‘Œπ‘˜π»ξ€Έξ€Έπ‘‡π‘†2=‖‖𝑅ξƒͺπ‘˜β€–β€–2βˆ’β€–β€–π‘…π‘˜β€–β€–2β€–β€–π‘ˆπ‘˜β€–β€–2+β€–β€–π‘‰π‘˜β€–β€–2ξƒ©π‘ˆtrπ‘‡π‘˜ξƒ¬π΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜,π‘Œπ‘˜π΅ξ€Έξ€Έπ‘‡+πΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜,π‘Œπ‘˜πΉξ€Έξ€Έπ‘‡2+π‘ƒπ΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜,π‘Œπ‘˜π΅ξ€Έξ€Έπ‘‡π‘„+π‘ƒπΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜,π‘Œπ‘˜πΉξ€Έξ€Έπ‘‡π‘„2ξƒ­+π‘‰π‘‡π‘˜ξƒ¬βˆ’πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜,π‘Œπ‘˜π·ξ€Έξ€Έπ‘‡βˆ’πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜,π‘Œπ‘˜π»ξ€Έξ€Έπ‘‡2+βˆ’π‘…πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜,π‘Œπ‘˜π·ξ€Έξ€Έπ‘‡π‘†βˆ’π‘…πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜,π‘Œπ‘˜π»ξ€Έξ€Έπ‘‡π‘†2=‖‖𝑅ξƒͺπ‘˜β€–β€–2βˆ’β€–β€–π‘…π‘˜β€–β€–2β€–β€–π‘ˆπ‘˜β€–β€–2+β€–β€–π‘‰π‘˜β€–β€–2ξ€·π‘ˆtrπ‘‡π‘˜π‘ˆπ‘˜+π‘‰π‘‡π‘˜π‘‰π‘˜ξ€Έξ€·π‘ˆ=0,trπ‘‡π‘˜+1π‘ˆπ‘˜ξ€Έξ€·π‘‰+trπ‘‡π‘˜+1π‘‰π‘˜ξ€Έπ΄=trξƒ©ξƒ¬π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜+1,π‘Œπ‘˜+1𝐡𝑇+πΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜+1,π‘Œπ‘˜+1𝐹𝑇2+π‘ƒπ΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜+1,π‘Œπ‘˜+1𝐡𝑇𝑄+π‘ƒπΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜+1,π‘Œπ‘˜+1𝐹𝑇𝑄2+β€–β€–π‘…π‘˜+1β€–β€–2β€–β€–π‘…π‘˜β€–β€–2π‘ˆπ‘˜ξƒ­π‘‡π‘ˆπ‘˜βŽžβŽŸβŽŸβŽ +trξƒ©ξƒ¬βˆ’πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜+1,π‘Œπ‘˜+1π·ξ€Έξ€Έπ‘‡βˆ’πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜+1,π‘Œπ‘˜+1𝐻𝑇2+βˆ’π‘…πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜+1,π‘Œπ‘˜+1π·ξ€Έξ€Έπ‘‡π‘†βˆ’π‘…πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜+1,π‘Œπ‘˜+1𝐻𝑇𝑆2+β€–β€–π‘…π‘˜+1β€–β€–2β€–β€–π‘…π‘˜β€–β€–2π‘‰π‘˜ξƒ­π‘‡π‘‰π‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽξƒ¬π΄=trπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜+1,π‘Œπ‘˜+1𝐡𝑇+πΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜+1,π‘Œπ‘˜+1𝐹𝑇+β€–β€–π‘…π‘˜+1β€–β€–2β€–β€–π‘…π‘˜β€–β€–2π‘ˆπ‘˜ξƒ­π‘‡π‘ˆπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽξƒ¬+trβˆ’πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜+1,π‘Œπ‘˜+1π·ξ€Έξ€Έπ‘‡βˆ’πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜+1,π‘Œπ‘˜+1𝐻𝑇+β€–β€–π‘…π‘˜+1β€–β€–2β€–β€–π‘…π‘˜β€–β€–2π‘‰π‘˜ξƒ­π‘‡π‘‰π‘˜βŽžβŽŸβŽŸβŽ ξ€·π‘ˆ=trπ‘‡π‘˜ξ€Ίπ΄π‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜+1,π‘Œπ‘˜+1𝐡𝑇+πΈπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜+1,π‘Œπ‘˜+1𝐹𝑇+π‘‰π‘‡π‘˜ξ€Ίβˆ’πΆπ‘‡ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜+1,π‘Œπ‘˜+1π·ξ€Έξ€Έπ‘‡βˆ’πΊπ‘‡ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜+1,π‘Œπ‘˜+1𝐻𝑇+β€–β€–π‘…ξ€»ξ€Έπ‘˜+1β€–β€–2β€–β€–π‘…π‘˜β€–β€–2ξ‚€β€–β€–π‘ˆπ‘˜β€–β€–2+β€–β€–π‘‰π‘˜β€–β€–2𝑋=trπ‘€βˆ’Ξ¦π‘˜+1,π‘Œπ‘˜+1ξ€Έξ€Έπ‘‡π΄π‘ˆπ‘˜ξ€·ξ€·π‘‹π΅+π‘βˆ’Ξ¨π‘˜+1,π‘Œπ‘˜+1ξ€Έξ€Έπ‘‡πΈπ‘ˆπ‘˜πΉβˆ’ξ€·ξ€·π‘‹π‘€βˆ’Ξ¦π‘˜+1,π‘Œπ‘˜+1ξ€Έξ€Έπ‘‡πΆπ‘‰π‘˜π·βˆ’ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜+1,π‘Œπ‘˜+1ξ€Έξ€Έπ‘‡πΊπ‘‰π‘˜π»ξ‚+β€–β€–π‘…π‘˜+1β€–β€–2β€–β€–π‘…π‘˜β€–β€–2ξ‚€β€–β€–π‘ˆπ‘˜β€–β€–2+β€–β€–π‘‰π‘˜β€–β€–2𝑋=trdiagπ‘€βˆ’Ξ¦π‘˜+1,π‘Œπ‘˜+1𝑇,ξ€·ξ€·π‘‹π‘βˆ’Ξ¨π‘˜+1,π‘Œπ‘˜+1ξ€Έξ€Έπ‘‡ξ‚ξ€·Ξ¦ξ€·π‘ˆdiagπ‘˜,π‘‰π‘˜ξ€Έξ€·π‘ˆ,Ξ¨π‘˜,π‘‰π‘˜ξ‚+β€–β€–π‘…ξ€Έξ€Έπ‘˜+1β€–β€–2β€–β€–π‘…π‘˜β€–β€–2ξ‚€β€–β€–π‘ˆπ‘˜β€–β€–2+β€–β€–π‘‰π‘˜β€–β€–2=β€–β€–π‘ˆπ‘˜β€–β€–2+β€–β€–π‘‰π‘˜β€–β€–2β€–β€–π‘…π‘˜β€–β€–2𝑅trπ‘‡π‘˜+1ξ€·π‘…π‘˜βˆ’π‘…π‘˜+1+β€–β€–π‘…ξ€Έξ€Έπ‘˜+1β€–β€–2β€–β€–π‘…π‘˜β€–β€–2ξ‚€β€–β€–π‘ˆπ‘˜β€–β€–2+β€–β€–π‘‰π‘˜β€–β€–2=0.(A.4) Hence, (A.2) holds for 𝑖=π‘˜. Therefore, (A.2) holds by the principle of induction.

Step 2. We show that 𝑅tr𝑇𝑖+1π‘…π‘—ξ€Έξ€·π‘ˆ=0,tr𝑇𝑖+1π‘ˆπ‘—+𝑉𝑇𝑖+1𝑉𝑗=0,𝑗=1,2,…,𝑖,βˆ€π‘–β‰₯1.(A.5)
When 𝑖=1, (A.5) holds.
Assume that 𝑅trπ‘‡π‘–π‘…π‘—ξ€Έξ€·π‘ˆ=0,trπ‘‡π‘–π‘ˆπ‘—+𝑉𝑇𝑖𝑉𝑗=0,𝑗=1,2,…,π‘ βˆ’1,βˆ€π‘ β‰₯2,(A.6) then we show that 𝑅tr𝑇𝑖+1π‘…π‘—ξ€Έξ€·π‘ˆ=0,tr𝑇𝑖+1π‘ˆπ‘—+𝑉𝑇𝑖+1𝑉𝑗=0,𝑗=1,2,…,𝑠.(A.7)
In fact, we have that 𝑅tr𝑇𝑖+1π‘…π‘—ξ€ΈβŽ›βŽœβŽœβŽ