Least Squares Based Iterative Algorithm for the Coupled Sylvester Matrix Equations

Abstract

By analyzing the eigenvalues of the related matrices, the convergence analysis of the least squares based iteration is given for solving the coupled Sylvester equations 𝐀𝐗+𝐘𝐁=𝐂 and 𝐃𝐗+𝐘𝐄=𝐅 in this paper. The analysis shows that the optimal convergence factor of this iterative algorithm is 1. In addition, the proposed iterative algorithm can solve the generalized Sylvester equation 𝐀𝐗𝐁+𝐂𝐗𝐃=𝐅. The analysis demonstrates that if the matrix equation has a unique solution then the least squares based iterative solution converges to the exact solution for any initial values. A numerical example illustrates the effectiveness of the proposed algorithm.

1. Introduction

Matrix equations arise in systems and control, such as Lyapunov matrix equation and Riccati equation. How to solve these matrix equations becomes a major field of the matrix computations [1–4]. Main points of this field are the decompositions and transformations of the matrices, eigenvalues and eigenvectors, and Krylov subspace [5–8]. Other points contain the algorithms and the convergence analysis. The algorithms provided a set of operation steps, and according to these steps it can find a solution of a matrix equation within finite steps in given error bounds [9–11]. The convergence analysis offered more details of an algorithm, and in general, these details indicated some new research areas [12–15].

The direct method and the indirect method are the two main approaches of solving the matrix equations [16–18]. With the development of the requirement of the computation and the matrix theory, the indirect method or iterative method becomes the main approach of solving the matrix equations [19, 20].

Iterative method is active in studying other engineering problems. Particularly, the iterative method can be used to identify systems [21–23] and estimate parameter of systems [24–28]. For example, iterative method is useful in the identification [29–32] and parameter estimation [33–36] of linear and nonlinear systems [37–46].

The Jacobi and Gauss-Seidel iterative methods were discussed in the literature. By extending the Jacobi and Gauss-Seidel iterative methods, Ding and his coworkers recently presented a large family of the least squares based iterative methods for solving the matrix equations 𝐀𝐱=𝐛 and 𝐀𝐗𝐁=𝐅 [47–49]. It has been proved that these least squares based iterative solutions always converge fast to the exact ones as long as the unique solutions exist. But the range of the convergence factor is still open. Motivated by the importance of this algorithm, we develop a new way to prove the convergence of the least squares based iterative algorithm for the coupled Sylvester matrix equations 𝐀𝐗+𝐘𝐁=𝐂 and 𝐃𝐗+𝐘𝐄=𝐅. According to the new proof, we obtain the optimal convergence factor of this iterative algorithm and extend the iterative algorithm to solve the generalized Sylvester matrix equation 𝐀𝐗𝐁+𝐂𝐗𝐃=𝐅. The algorithm in this paper can be extended to other more general and complex matrix equations [50–52].

The paper is organized as follows. Section 2 gives some preliminaries. Section 3 presents a new proof to the least squares iterative algorithm for solving the coupled Sylvester matrix equations 𝐀𝐗+𝐘𝐁=𝐂 and 𝐃𝐗+𝐘𝐄=𝐅. Section 4 revises this algorithm to solve the equation 𝐀𝐗𝐁+𝐂𝐗𝐃=𝐅. Section 5 gives an example to illustrate the effectiveness of the proposed results. Finally, we offer some concluding remarks in Section 6.

2. Basic Preliminaries

Some symbols and lemmas are introduced first. 𝐈_𝑛 is the identity matrix of size 𝑛×𝑛. 𝐈 is an identity matrix with an appropriate size. 𝜆[𝐌] denotes the eigenvalue set of matrix 𝐌. |𝐗|=det[𝐗] denotes the matrix determinant. ‖𝐗‖ denotes the Frobenius norm of 𝐗 and is defined as ‖𝐗‖²∶=tr[𝐗𝐗^𝑇]. For an 𝑚×𝑛 matrix 𝐗=𝐱₁,𝐱₂,…,𝐱_𝑛∈ℝ^𝑚×𝑛,𝐱_𝑖∈ℝ^𝑚; (1)col[𝐗] denotes an 𝑚𝑛-dimensional vector formed by (col[𝐗])^𝑇=𝐱^𝑇₁,𝐱^𝑇₂,…,𝐱^𝑇_𝑛∈ℝ^𝑚𝑛. (2)𝐀⊗𝐁 represents the Kronecker product of matrices 𝐀 and 𝐁. A formula involved the vec-operator col and the Kronecker product is col[𝐀𝐗𝐁]=(𝐁^𝑇⊗𝐀)col[𝐗] [53].

If matrix 𝐀^𝑇𝐀 is invertible, then 𝐀(𝐀^𝑇𝐀)⁻¹𝐀^𝑇 is symmetric and idempotent. To be more specific, we have the following lemma.

Lemma 1. For the symmetric matrix 𝐀(𝐀^𝑇𝐀)⁻¹𝐀^𝑇, there exists an orthogonal matrix 𝐐 such that 𝐐^𝑇𝐀𝐀^𝑇𝐀⁻¹𝐀^𝑇𝐐=diag[1,…,1,0,…,0]=∶Λ. (3) Moreover, rank[𝐀(𝐀^𝑇𝐀)⁻¹𝐀^𝑇]=rank[Λ]=𝑛.

3. The Coupled Sylvester Matrix Equations

In this section, we will prove the convergence of the least squares iterative algorithm for solving the following coupled Sylvester matrix equations: 𝐀𝐗+𝐘𝐁=𝐂,𝐃𝐗+𝐘𝐄=𝐅, (4) where 𝐀,𝐃∈ℝ^𝑚×𝑚, 𝐁,𝐄∈ℝ^𝑛×𝑛, and 𝐂,𝐅∈ℝ^𝑚×𝑛 are known and 𝐗,𝐘∈ℝ^𝑚×𝑛 are to be determined. By using the hierarchical identification principle, the least squares based iterative algorithm for solving (4) is given by [47] 𝐗(𝑘)=𝐗(𝑘−1)+𝜇𝐀^𝑇𝐀+𝐃^𝑇𝐃⁻¹𝐀𝐃^𝑇×()()()(),𝐘(𝑘)()[()()()()𝐄]×[]𝐂−𝐘𝑘−1𝐁−𝐀𝐗𝑘−1𝐅−𝐘𝑘−1𝐄−𝐃𝐗𝑘−1=𝐘𝑘−1+𝜇𝐂−𝐀𝐗𝑘−1−𝐘𝑘−1𝐁,𝐅−𝐃𝐗𝑘−1−𝐘𝑘−1𝐁,𝐄^𝑇𝐁𝐁^𝑇+𝐄𝐄^𝑇−1,20<𝜇<𝜆_max𝐍⁻¹𝐌. (5) Here, 𝐍=diag[𝐈_𝑛⊗(𝐀^𝑇𝐀+𝐃^𝑇𝐃),(𝐁𝐁^𝑇+𝐄𝐄^𝑇)⊗𝐈_𝑚] and 𝐈𝐌=^𝑛⊗𝐀^𝑇𝐀+𝐃^𝑇𝐃𝐁^𝑇⊗𝐀^𝑇+𝐄^𝑇⊗𝐃^𝑇𝐁⊗𝐀+𝐄⊗𝐃𝐁𝐁^𝑇+𝐄𝐄^𝑇⊗𝐈_𝑚. (6) To initialize the algorithm, we take 𝐗(0) and 𝐘(0) as some small real matrices, for example, 𝐗(0)=𝐘(0)=10⁻⁶𝟏_𝑚×𝑛 with 𝟏_𝑚×𝑛 being an 𝑚×𝑛 matrix whose elements are all 1.

Theorem 2. If (4) has unique solutions 𝐗 and 𝐘, then for any initial values 𝐗(0) and 𝐘(0) the iterative solutions 𝐗(𝑘) and 𝐘(𝑘) given by iteration in (5) converge to 𝐗 and 𝐘; that is, lim_𝑘→∞𝐗(𝑘)=𝐗,lim_𝑘→∞𝐘(𝑘)=𝐘; (7) or the error matrices 𝐗(𝑘)−𝐗 and 𝐘(𝑘)−𝐘 converge to zero; in this case, the optimal convergence factor is 𝜇=1.

Proof. Define two error matrices 𝐗(𝑘)∶=𝐗(𝑘)−𝐗,𝐘(𝑘)∶=𝐘(𝑘)−𝐘. (8) Using (4) and (5) gives 𝐗(𝑘)=𝐗(𝑘−1)−𝜇𝐀^𝑇𝐀+𝐃^𝑇𝐃⁻¹𝐀𝐃^𝑇×𝐘()𝐗()𝐘()𝐗(),𝐘(𝑘)=𝐘()𝐘()𝐗(),𝐘()𝐗(×[]𝑘−1𝐁+𝐀𝑘−1𝑘−1𝐄+𝐃𝑘−1𝑘−1−𝜇𝑘−1𝐁+𝐀𝑘−1𝑘−1𝐄+𝐃𝑘−1)𝐁,𝐄^𝑇𝐁𝐁^𝑇+𝐄𝐄^𝑇−1. (9) Expanding (9) gives 𝐗(𝑘)=𝐗(𝑘−1)−𝜇𝐀^𝑇𝐀+𝐃^𝑇𝐃⁻¹×𝐀^𝑇𝐘(𝑘−1)𝐁+𝐀^𝑇𝐀𝐗(𝑘−1)+𝐃^𝑇𝐘(𝑘−1)𝐄+𝐃^𝑇𝐃𝐗(,𝐘(𝑘)=𝐘()𝐘()𝑘−1)𝑘−1−𝜇𝑘−1𝐁𝐁^𝑇+𝐀𝐗(𝑘−1)𝐁^𝑇+𝐘(𝑘−1)𝐄𝐄^𝑇+𝐃𝐗(𝑘−1)𝐄^𝑇×𝐁𝐁^𝑇+𝐄𝐄^𝑇−1. (10) Simplifying (10) gives 𝐗(𝑘)=1−𝜇𝐗(𝑘−1)−𝜇𝐀^𝑇𝐀+𝐃^𝑇𝐃⁻¹×𝐀^𝑇𝐘(𝑘−1)𝐁+𝐃^𝑇𝐘()𝐄,𝐘(𝑘)=𝐘()𝐀𝐗()𝐁𝑘−11−𝜇𝑘−1−𝜇𝑘−1^𝑇+𝐃𝐗(𝑘−1)𝐄^𝑇×𝐁𝐁^𝑇+𝐄𝐄^𝑇−1. (11) Using the formula col[𝐀𝐗𝐁]=(𝐁^𝑇⊗𝐀)col[𝐗] gives 𝐗(𝑘=𝐗(𝐈col)1−𝜇col𝑘−1)−𝜇_𝑛⊗𝐀^𝑇𝐀+𝐃^𝑇𝐃⁻¹×𝐁^𝑇⊗𝐀^𝑇+𝐄^𝑇⊗𝐃^𝑇𝐘(,𝐘(𝑘=𝐘(×col𝑘−1)col)1−𝜇col𝑘−1)−𝜇𝐁𝐁^𝑇+𝐄𝐄^𝑇−1⊗𝐈_𝑚×[𝐁⊗𝐀+𝐄⊗𝐃]col𝐗(𝑘−1). (12) Set 𝐉∶=𝐁⊗𝐈_𝑚,𝐄⊗𝐈_𝑚∈ℝ^{(𝑚𝑛)×(2𝑚𝑛)},𝐈𝐊∶=^𝑛𝐈⊗𝐀_𝑛⊗𝐃∈ℝ^{(2𝑚𝑛)×(𝑚𝑛)},𝐈𝐋∶=^𝑚𝑛𝐊^𝑇𝐊⁻¹(𝐉𝐊)^𝑇𝐉𝐉^𝑇−1(𝐉𝐊)𝐈_𝑚𝑛. (13) Using these symbols, a compact form of (12) is 𝐗(𝑘),𝐘(𝑘=𝐈col)^𝑚𝑛𝟎𝟎𝐈_𝑚𝑛𝐗(),𝐘(.−𝜇𝐋×col𝑘−1𝑘−1) (14) Set 𝐈𝐌∶=^𝑚𝑛𝟎𝟎𝐈_𝑚𝑛−𝜇𝐋. (15) The eigenvalues of 𝐌 inside the unit circle will complete the proof. Let 𝛼 be an eigenvalue of 𝐌; we will show that |𝛼|<1, or, if 𝛽∈𝜆[𝐋], then 0<𝛽<2.
Consider the following characteristic polynomial of 𝐋: 𝑝(𝑧)=𝑧𝐈_2𝑚𝑛=−𝐋𝑧𝐈_𝑚𝑛𝟎𝟎𝑧𝐈_𝑚𝑛−𝐈_𝑚𝑛𝐊^𝑇𝐊⁻¹(𝐉𝐊)^𝑇𝐉𝐉^𝑇−1(𝐉𝐊)𝐈_𝑚𝑛=(𝑧−1)𝐈_𝑚𝑛−𝐊^𝑇𝐊⁻¹(𝐉𝐊)^𝑇−𝐉𝐉^𝑇−1(𝐉𝐊)(𝑧−1)𝐈_𝑚𝑛. (16) It is not hard to show that 𝑝(2)≠0 by calculation. Since 𝑝(2)=𝐈_𝑚𝑛−𝐊^𝑇𝐊⁻¹(𝐉𝐊)^𝑇−𝐉𝐉^𝑇−1(𝐉𝐊)𝐈_𝑚𝑛=𝐈_𝑚𝑛−𝐊^𝑇𝐊⁻¹(𝐉𝐊)^𝑇−𝐉𝐉^𝑇−1(𝐉𝐊)𝐈_𝑚𝑛=𝐊^𝑇𝐊⁻¹𝟎𝟎𝐉𝐉^𝑇−1𝐊^𝑇𝐊−(𝐉𝐊)^𝑇−(𝐉𝐊)𝐉𝐉^𝑇=𝐊^𝑇𝐊⁻¹𝟎𝟎𝐉𝐉^𝑇−1𝐊^𝑇−𝐉𝐊,−𝐉^𝑇, (17) it follows from the formula =|𝐀𝐀𝐁𝐂𝐃|𝐃−𝐂𝐀⁻¹𝐁 (18) that 𝐊,−𝐉^𝑇=𝐈_𝑛⊗𝐀−𝐁^𝑇⊗𝐈_𝑚𝐈_𝑛⊗𝐃−𝐄^𝑇⊗𝐈_𝑚=𝐈_𝑛×⊗𝐀−𝐄^𝑇⊗𝐈_𝑚+𝐈_𝑛⊗𝐃𝐈_𝑛⊗𝐀⁻¹𝐁^𝑇⊗𝐈_𝑚=𝐈_𝑛⊗𝐀−𝐄^𝑇⊗𝐈_𝑚+𝐁^𝑇⊗𝐃𝐀⁻¹=−𝐄^𝑇⊗𝐈_𝑚+𝐁^𝑇⊗𝐃𝐀⁻¹𝐈_𝑛=⊗𝐀−𝐄^𝑇⊗𝐈_𝑚+𝐁^𝑇⊗𝐃𝐀⁻¹𝐈_𝑛=⊗𝐀−𝐄^𝑇⊗𝐀+𝐁^𝑇⊗𝐃. (19) On the other hand, since (4) has a unique solution, it follows that the matrix 𝐈_𝑛⊗𝐀𝐁^𝑇⊗𝐈_𝑚𝐈_𝑛⊗𝐃𝐄^𝑇⊗𝐈_𝑚 (20) is invertible. A determinant expansion shows that 𝐈_𝑛⊗𝐀𝐁^𝑇⊗𝐈_𝑚𝐈_𝑛⊗𝐃𝐄^𝑇⊗𝐈_𝑚=𝐈_𝑛×𝐄⊗𝐀_𝑇⊗𝐈_𝑚−𝐈_𝑛⊗𝐃𝐈_𝑛⊗𝐀⁻¹𝐁^𝑇⊗𝐈_𝑚=𝐈_𝑛⊗𝐀𝐄^𝑇⊗𝐈_𝑚−𝐁^𝑇⊗𝐃𝐀⁻¹=𝐄^𝑇⊗𝐈_𝑚−𝐁^𝑇⊗𝐃𝐀⁻¹𝐈_𝑛=𝐄⊗𝐀_𝑇⊗𝐈_𝑚−𝐁^𝑇⊗𝐃𝐀⁻¹𝐈_𝑛=𝐄⊗𝐀_𝑇⊗𝐀−𝐁^𝑇=()⊗𝐃−1^𝑚𝑛−𝐄^𝑇⊗𝐀+𝐁^𝑇=()⊗𝐃−1^𝑚𝑛𝐊,−𝐉^𝑇≠0. (21) So it gives 𝑝(2)≠0. Since 𝑝(𝑧)=(𝑧−1)𝐈_𝑚𝑛−𝐊^𝑇𝐊⁻¹(𝐉𝐊)^𝑇−𝐉𝐉^𝑇−1(𝐉𝐊)(𝑧−1)𝐈_𝑚𝑛, (22) expanding it gives 𝑝(𝑧)=(𝑧−1)𝐈_𝑚𝑛(𝑧−1)𝐈_𝑚𝑛−𝐉𝐉^𝑇−1()×()𝐉𝐊𝑧−1⁻¹𝐈_𝑚𝑛𝐊^𝑇𝐊⁻¹(𝐉𝐊)^𝑇=(𝑧−1)²𝐈_𝑚𝑛−𝐉𝐉^𝑇−1(𝐉𝐊)𝐊^𝑇𝐊⁻¹(𝐉𝐊)^𝑇. (23) Setting 𝐏∶=(𝐉𝐉^𝑇)⁻¹(𝐉𝐊)(𝐊^𝑇𝐊)⁻¹(𝐉𝐊)^𝑇∈ℝ^{(𝑚𝑛)×(𝑚𝑛)}, it can be rewritten as 𝐏=𝐉𝐉^𝑇−1𝐉𝐊𝐊^𝑇𝐊⁻¹𝐊^𝑇𝐉^𝑇. (24) According to Lemma 1, there exists an orthogonal matrix 𝐐∶=[𝐪₁,𝐪₂,…,𝐪_2𝑚𝑛] such that 𝐐^𝑇𝐊𝐊^𝑇𝐊⁻¹𝐊^𝑇[],[𝐐𝐐=diag1,1,…,1,0,0,…,0rankdiag1,1,…,1,0,0,…,0]=rank^𝑇𝐊𝐊^𝑇𝐊⁻¹𝐊^𝑇𝐐=rank𝐊𝐊^𝑇𝐊⁻¹𝐊^𝑇=rank[𝐊]=𝑚𝑛. (25) Using (25) can be manipulated to get 𝐊𝐊^𝑇𝐊⁻¹𝐊^𝑇𝐪₁,𝐪₂,…,𝐪_2𝑚𝑛=𝐪₁,𝐪₂,…,𝐪_2𝑚𝑛[]=𝐪diag1,1,…,1,0,0,…,0₁,𝐪₂,…,𝐪_𝑚𝑛,𝟎,𝟎,…,𝟎. (26) If 𝜌∈𝜆[𝐏], then there exist the nonzero vectors 𝐲∈ℝ^𝑚𝑛 that satisfy 𝐏𝐲=𝜌𝐲, which can be written as 𝐉𝐉^𝑇−1𝐉𝐊𝐊^𝑇𝐊⁻¹𝐊^𝑇𝐉^𝑇𝐊𝐲=𝜌𝐲,𝐉𝐊^𝑇𝐊⁻¹𝐊^𝑇𝐉^𝑇𝐲=𝜌𝐉𝐉^𝑇𝐲. (27) Since 𝐉^𝑇𝐲∈ℝ^2𝑚𝑛 and 𝐐∈ℝ^{(2𝑚𝑛)×(2𝑚𝑛)} are orthogonal matrices, it follows that 𝐉^𝑇𝐲=𝑘₁𝐪₁+𝑘₂𝐪₂+⋯+𝑘_2𝑚𝑛𝐪_2𝑚𝑛, (28) where 𝑘_𝑖∈ℝ,𝑖=1,2,…,2𝑚𝑛 and 𝑘²₁+𝑘²₂+⋯+𝑘²_2𝑚𝑛≠0. Then from (28), it gives 𝐉^𝑇𝐲^𝑇𝐉^𝑇𝐲=𝑘₁𝐪₁+𝑘₂𝐪₂+⋯+𝑘_2𝑚𝑛𝐪_2𝑚𝑛^𝑇×𝑘₁𝐪₁+𝑘₂𝐪₂+⋯+𝑘_2𝑚𝑛𝐪_2𝑚𝑛=𝑘²₁+𝑘²₂+⋯+𝑘²_2𝑚𝑛. (29) Using (26) gives 𝐊𝐊^𝑇𝐊⁻¹𝐊^𝑇𝐉^𝑇𝐲=𝐊𝐊^𝑇𝐊⁻¹𝐊^𝑇𝑘₁𝐪₁+𝑘₂𝐪₂+⋯+𝑘_2𝑚𝑛𝐪_2𝑚𝑛=𝑘₁𝐪₁+𝑘₂𝐪₂+⋯+𝑘_𝑚𝑛𝐪_𝑚𝑛. (30) According to (29) and (30), (27) can be manipulated to get 𝐲^𝑇𝐉𝐊𝐊^𝑇𝐊⁻¹𝐊^𝑇𝐉^𝑇𝐲=𝐲^𝑇𝜌𝐉𝐉^𝑇𝐲𝐲,_𝑇𝐉𝑘₁𝐪₁+𝑘₂𝐪₂+⋯+𝑘_𝑚𝑛𝐪_𝑚𝑛=𝜌𝑘²₁+𝑘²₂+⋯+𝑘²_2𝑚𝑛,𝑘²₁+𝑘²₂+⋯+𝑘²_𝑚𝑛=𝜌𝑘²₁+𝑘²₂+⋯+𝑘²_2𝑚𝑛. (31) So it gives 𝑘0⩽𝜌=²¹+𝑘²₂+⋯+𝑘²_𝑚𝑛𝑘²₁+𝑘²₂+⋯+𝑘²_2𝑚𝑛⩽1. (32) Suppose that 𝜆[𝐏]={𝜌²₁,𝜌²₂,…,𝜌²_𝑚𝑛}, where 0⩽𝜌_𝑖⩽1, 𝑖=1,2,…,𝑚𝑛. From Schur decomposition theorem, there exists a decomposition 𝐔^𝑇𝐏𝐔=𝐃+𝐅, where 𝐔 is an orthogonal matrix, 𝐃=diag(𝜌²₁,𝜌²₂,…,𝜌²_𝑚𝑛), and 𝐅 is a strictly upper triangular matrix. Then from (23), it gives 𝑝(𝑧)=(𝑧−1)²𝐈_𝑚𝑛−𝐉𝐉^𝑇−1(𝐉𝐊)𝐊^𝑇𝐊⁻¹(𝐉𝐊)^𝑇=(𝑧−1)²𝐈_𝑚𝑛=𝐔−𝐏_𝑇(𝑧−1)²𝐈_𝑚𝑛𝐔|=𝐔−𝐏|_𝑇(𝑧−1)²𝐈_𝑚𝑛𝐔−𝐔^𝑇=)𝐏𝐔(𝑧−1²𝐈_𝑚𝑛=)−𝐃−𝐅(𝑧−1²−𝜌²₁(𝑧−1)²−𝜌²₂⋯(𝑧−1)²−𝜌²_𝑚𝑛=𝑧−1−𝜌₁𝑧−1+𝜌₁𝑧−1−𝜌₂𝑧−1+𝜌₂⋯𝑧−1−𝜌_𝑚𝑛𝑧−1+𝜌_𝑚𝑛. (33) It follows that 𝜆[𝐋]={1+𝜌₁,1−𝜌₁,1+𝜌₂,1−𝜌₂,…,1+𝜌_𝑚𝑛,1−𝜌_𝑚𝑛}. Since 𝑝(2)≠0, we obtain 𝜌_𝑖<1, 𝑖=1,2,…,𝑚𝑛. Thus, 0<𝛽<2.
Next, we determine the optimal convergence factor. From 𝜆[𝐋]={1+𝜌₁,1−𝜌₁,1+𝜌₂,1−𝜌₂,…,1+𝜌_𝑚𝑛,1−𝜌_𝑚𝑛} and 𝐈𝐌=^𝑚𝑛𝟎𝟎𝐈_𝑚𝑛−𝜇𝐋, (34) one gets 𝜆[𝐌]={1−𝜇(1+𝜌₁),1−𝜇(1−𝜌₁),…,1−𝜇(1+𝜌_𝑚𝑛),1−𝜇(1−𝜌_𝑚𝑛)}. Set 𝜌₁∶=max{𝜌₁,𝜌₂,…,𝜌_𝑚𝑛}. Taking absolute values of these eigenvalues, the optimal convergence factor satisfies minmax1−𝜇1+𝜌₁,1−𝜇1−𝜌₁,…,1−𝜇1+𝜌_𝑚𝑛,1−𝜇1−𝜌_𝑚𝑛=minmax1−𝜇1+𝜌₁,1−𝜇1−𝜌₁. (35) Equation (35) is equivalent to |1−𝜇(1+𝜌₁)|=|1−𝜇(1−𝜌₁)|. Solving it gives 𝜇=1. The proof is completed.

4. The Generalized Sylvester Matrix Equation AXB + CXD = F

In this section, we use iteration in (5) to solve the generalized Sylvester matrix equation. Consider the following equation: 𝐀𝐗𝐁+𝐂𝐗𝐃=𝐅, (36) where 𝐀,𝐂∈ℝ^𝑚×𝑚, 𝐁,𝐃∈ℝ^𝑛×𝑛, and 𝐅∈ℝ^𝑚×𝑛 are given constant matrices and 𝐗∈ℝ^𝑚×𝑛 is the unknown matrix to be solved. The following conclusion is obvious.

Equation (36) has a unique solution if and only if rank𝐁^𝑇⊗𝐀+𝐃^𝑇[𝐅𝐁⊗𝐂,col]=rank^𝑇⊗𝐀+𝐃^𝑇⊗𝐂=𝑚𝑛. (37) In this case, the unique solution is given by col[𝐗]=(𝐁^𝑇⊗𝐀+𝐃^𝑇⊗𝐂)⁻¹col[𝐅].

Setting 𝐕∶=𝐗𝐁∈ℝ^𝑚×𝑛 and 𝐖∶=𝐂𝐗∈ℝ^𝑚×𝑛, (36) can be equivalently expressed as 𝐀𝐕+𝐖𝐃=𝐅,𝐂𝐕−𝐖𝐁=𝟎. (38) If 𝐈rank^𝑛⊗𝐀𝐃^𝑇⊗𝐈_𝑚𝐈_𝑛⊗𝐂−𝐁^𝑇⊗𝐈_𝑚=2𝑚𝑛, (39) then (38) has a unique solution. It is easy to show that if |𝐂|≠0 or |𝐁|≠0, then (38) is equivalent to (36). Next, we show that if |𝐂|≠0 or |𝐁|≠0, then (39) is equivalent to (37). That is, we have the following determinant result: 𝐈_𝑛⊗𝐂−𝐁^𝑇⊗𝐈_𝑚𝐈_𝑛⊗𝐀𝐃^𝑇⊗𝐈_𝑚=𝐈_𝑛⊗𝐂𝐃^𝑇⊗𝐈_𝑚+𝐈_𝑛⊗𝐀𝐈_𝑛⊗𝐂⁻¹𝐁^𝑇⊗𝐈_𝑚=𝐈_𝑛⊗𝐂𝐃^𝑇⊗𝐈_𝑚+𝐁^𝑇⊗𝐀𝐂⁻¹=𝐃^𝑇⊗𝐈_𝑚+𝐁^𝑇⊗𝐀𝐂⁻¹𝐈_𝑛=𝐃⊗𝐂_𝑇⊗𝐂+𝐁^𝑇⊗𝐀. (40) According to Theorem 2, (38) can be solved by iteration in (5), and from 𝐗𝐁=𝐕 or 𝐂𝐗=𝐖, (36) can be solved.

5. Example

In this section, an example is offered to illustrate the convergence of the proposed iterative algorithm.

Example 1. Consider the coupled Sylvester matrix equations in the form of (4) with ,,.𝐀=7−5−19−8,𝐁=−112−8−14𝐂=−1213−259−215,𝐃=−102116𝐄=−10−3−1−12,𝐅=−104−9041−20 (41) The unique solution is found to be 𝑥𝐗=¹¹𝑥₁₂𝑥₂₁𝑥₂₂=45𝑦94,𝐘=¹¹𝑦₁₂𝑦₂₁𝑦₂₂=.8257 (42)
Taking 𝐗(0)=𝐘(0)=10⁻⁶𝟏_2×2 as the initial iterative values and using iteration (5) to compute 𝐗(𝑘) and 𝐘(𝑘), the iterative values of 𝐗(𝑘) and 𝐘(𝑘) are shown in Table 1 with the relative error 𝛿∶=‖𝐗(𝑘)−𝐗‖²+‖𝐘(𝑘)−𝐘‖²‖𝐗‖²+‖𝐘‖². (43) The effect of changing the convergence factor 𝜇 is illustrated in Figure 1.

Table 1 

The least squares iterative solutions (𝜇 = 1.0).

Figure 1 

The relative error 𝛿 versus iteration 𝑘.

From Table 1 and Figure 1, we find that the relative error goes to zero with the increasing of the iterative times. This shows that the proposed iterative algorithm is effective. In addition, Figure 1 shows that the optimal convergence factor 𝜇=1.0. This indicates that the result of the optimal convergence suggested in this paper is correct.

6. Conclusions

This paper proved the convergence of the least squares based iterative algorithm of the coupled Sylvester matrix equations 𝐀𝐗+𝐘𝐁=𝐂 and 𝐃𝐗+𝐘𝐄=𝐅, and the proof determined the range of the convergence factor and the optimal convergence factor. The suggested algorithm can also be used to solve the generalized Sylvester equation 𝐀𝐗𝐁+𝐂𝐗𝐃=𝐅. An example indicated that the iterative solution given by the least squares based iterative algorithm converges fast to its exact solution under proper conditions.

Conflict of Interests

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

Acknowledgment

This work was supported by the National Natural Science Foundation of China (no. 6110218).