Finite Iterative Algorithm for Solving a Complex of Conjugate and Transpose Matrix Equation

Abstract

We consider an iterative algorithm for solving a complex matrix equation with conjugate and transpose of two unknowns of the form: 𝐴₁𝑉𝐵₁+𝐶₁𝑊𝐷₁+𝐴₂𝑉𝐵₂+𝐶₂𝑊𝐷₂+𝐴₃𝑉^𝐻𝐵₃+𝐶₃𝑊^𝐻𝐷₃+𝐴₄𝑉^𝑇𝐵₄ + 𝐶₄𝑊^𝑇𝐷₄=𝐸. With the iterative algorithm, the existence of a solution of this matrix equation can be determined automatically. When this matrix equation is consistent, for any initial matrices 𝑉₁, 𝑊₁ the solutions can be obtained by iterative algorithm within finite iterative steps in the absence of round-off errors. Some lemmas and theorems are stated and proved where the iterative solutions are obtained. A numerical example is given to illustrate the effectiveness of the proposed method and to support the theoretical results of this paper.

1. Introduction

Consider the complex matrix equation: 𝐴₁𝑉𝐵₁+𝐶₁𝑊𝐷₁+𝐴₂𝑉𝐵₂+𝐶₂𝑊𝐷₂+𝐴₃𝑉^𝐻𝐵₃+𝐶₃𝑊^𝐻𝐷₃+𝐴₄𝑉^𝑇𝐵₄+𝐶₄𝑊^𝑇𝐷₄=𝐸, (1) where 𝐴₁,𝐴₂,𝐶₁,𝐶₂∈ℂ^𝑚×𝑟, 𝐵₁,𝐵₂,𝐷₁,𝐷₂∈ℂ^𝑠×𝑛, 𝐴₃,𝐴₄,𝐶₃,𝐶₄∈ℂ^𝑚×𝑠, 𝐸∈ℂ^𝑚×𝑛 and 𝐵₃,𝐵₄,𝐷₃,𝐷₄∈ℂ^𝑟×𝑛 are given matrices, while 𝑉,𝑊∈ℂ^𝑟×𝑠 are matrices to be determined. In the field of linear algebra, iterative algorithms for solving matrix equations have received much attention. Based on the iterative solutions of matrix equations, Ding and Chen presented the hierarchical gradient iterative algorithms for general matrix equations [1, 2] and hierarchical least squares iterative algorithms for generalized coupled Sylvester matrix equations and general coupled matrix equations [3, 4]. The hierarchical gradient iterative algorithms [1, 2] and hierarchical least squares iterative algorithms [1, 4, 5] for solving general (coupled) matrix equations are innovational and computationally efficient numerical ones and were proposed based on the hierarchical identification principle [3, 6] which regards the unknown matrix as the system parameter matrix to be identified. Iterative algorithms were proposed for continuous and discrete Lyapunov matrix equations by applying the hierarchical identification principle[7]. Recently, the idea of the hierarchical identification was also utilized to solve the so-called extended Sylvester-conjugate matrix equations in [8]. From an optimization point of view, a gradient-based iteration was constructed in [9] to solve the general coupled matrix equation. A significant feature of the method in [9] is that a necessary and sufficient condition guaranteeing the convergence of the algorithm can be explicitly obtained.

Some complex matrix equations have attracted attention from many researchers since it was shown in [10] that the consistence of the matrix equation 𝐴𝑋−𝑋𝐵=𝐶 can be characterized by the consimilarity [11–13] of two partitioned matrices related to the coefficient matrices 𝐴,𝐵, and 𝐶. By consimilarity Jordan decomposition, explicit solutions were obtained in [10, 14]. Some explicit expressions of the solution to the matrix equation 𝐴𝑋−𝑋𝐵=𝐶 were established in [15], and it was shown that this matrix equation has a unique solution if and only if 𝐴𝐴 and 𝐵𝐵 hav no common eigenvalues. Research on solving linear matrix equations has been actively engaged in for many years. For example, Navarra et al. studied a representation of the general common solution of the matrix equations 𝐴₁𝑋𝐵₁=𝐶₁ and 𝐴₂𝑋𝐵₂=𝐶₂ [16]; Van der Woude obtained the existence of a common solution 𝑋 for the matrix equations 𝐴_𝑖𝑋𝐵_𝑗=𝐶_𝑖𝑗 [17]; Bhimasankaram considered the linear matrix equations 𝐴𝑋=𝐵,𝐶𝑋=𝐷, and 𝐸𝑋𝐹=𝐺 [18]. Mitra has provided conditions for the existence of a solution and a representation of the general common solution of the matrix equations 𝐴𝑋=𝐶 and 𝑋𝐵=𝐷 and the matrix equations 𝐴₁𝑋𝐵₁=𝐶₁ and 𝐴₂𝑋𝐵₂=𝐶₂ [19, 20]. Ramadan et al. [21] introduced a complete, general, and explicit solution to the Yakubovich matrix equation 𝑉−𝐴𝑉𝐹=𝐵𝑊, and the matrix equation (𝐴𝑋𝐵,𝐺𝑋𝐻)=(𝐶,𝐷) has some important results that have been developed. In [22], necessary and sufficient conditions for its solvability and the expression of the solution were derived by means of generalized inverse. Moreover, in [22] the least-squares solution was also obtained by using the generalized singular value decomposition. While in [23], when this matrix equation is consistent, the minimum-norm solution was given by the use of the canonical correlation decomposition. In [24], based on the projection theorem in Hilbert space, an analytical expression of the least-squares solution was given for the matrix equation (𝐴𝑋𝐵,𝐺𝑋𝐻)=(𝐶,𝐷) by making use of the generalized singular value decomposition and the canonical correlation decomposition. In [25], by using the matrix rank method a necessary and sufficient condition was derived for the matrix equations 𝐴𝑋₁𝐵=𝐶 and 𝐺𝑋₂𝐻=𝐷 to have a common least square solution. In the aforementioned methods, the coefficient matrices of the considered equations are required to be firstly transformed into some canonical forms. Recently, an iterative algorithm has presented in [26] to solve the matrix equation (𝐴𝑋𝐵,𝐺𝑋𝐻)=(𝐶,𝐷). Different from the above mentioned methods, this algorithm can be implemented by initial coefficient matrices and can provide a solution within finite iteration steps for any initial values.

Very recently, in [27] a new operator of conjugate product for complex polynomial matrices is proposed. It is shown that an arbitrary complex polynomial matrix can be converted into the so-called Smith normal form by elementary transformations in the framework of conjugate product. Meanwhile, the conjugate product and the Sylvester-conjugate sum are also proposed in [28]. Based on the important properties of the above new operators, a unified approach to solve a general class of Sylvester-polynomial-conjugate matrix equations is given. The complete solution of the Sylvester-polynomial-conjugate matrix equation is obtained. In [29] by using a real inner product in complex matrix spaces, a solution can be obtained within finite iterative steps for any initial values in the absence of round-off errors. In [30] iterative solutions to a class of complex matrix equations are given by applying the hierarchical identification principle.

This paper is organized as follows. First, in Section 2, we introduce some notations, a lemma, and a theorem that will be needed to develop this work. In Section 3, we propose iterative methods to obtain numerical solution to the complex matrix equation with conjugate and transpose of two unknowns of the form: 𝐴₁𝑉𝐵₁+𝐶₁𝑊𝐷₁+𝐴₂𝑉𝐵₂+𝐶₂𝑊𝐷₂+𝐴₃𝑉^𝐻𝐵₃+𝐶₃𝑊^𝐻𝐷₃+𝐴₄𝑉^𝑇𝐵₄+𝐶₄𝑊^𝑇𝐷₄=𝐸 using iterative method. In Section 4, numerical example is given to explore the simplicity and the neatness of the presented methods.

2. Preliminaries

The following notations, definitions, lemmas, and theorems will be used to develop the proposed work. We use 𝐴^𝑇,𝐴,𝐴^𝐻, tr(𝐴), and ‖𝐴‖ to denote the transpose, conjugate, conjugate transpose, the trace, and the Frobenius norm of a matrix 𝐴, respectively. We denote the set of all 𝑚×𝑛 complex matrices by ℂ^𝑚×𝑛, and Re(𝑎) denote the real part of number 𝑎.

Definition 1 (inner product [31]). A real inner product space is a vector space 𝑉 over the real field ℝ together with an inner product. That is, with a map ⟨⋅,⋅⟩∶𝑉×𝑉⟶ℝ. (2) Satisfying the following three axioms for all vectors 𝑥,𝑦,𝑧∈𝑉 and all scalars 𝑎∈ℝ: (1)symmetry: ⟨𝑥,𝑦⟩=⟨𝑦,𝑥⟩(2)linearity in the first argument: ,=⟨⟩+𝑎𝑥,𝑦=𝑎𝑥,𝑦𝑥+𝑦,𝑧𝑥,𝑧𝑦,𝑧 (3)(3)positive definiteness: ⟨𝑥,𝑥⟩> 0 for all 𝑥≠0,
two vectors 𝑢,𝑣∈𝑉 are said to be orthogonal if ⟨𝑢,𝑣⟩=0.
The following theorem defines a real inner product on space ℂ^𝑚×𝑛 over the field ℝ.

Theorem 2 (see [32]). In the space ℂ^𝑚×𝑛 over the field ℝ, an inner product can be defined as ⟨𝐴,𝐵⟩=Retr𝐴^𝐻𝐵. (4)

3. The Main Result

In this section, we propose an iterative solution to a complex matrix equation with conjugate and transpose of two unknowns: 𝐴₁𝑉𝐵₁+𝐶₁𝑊𝐷₁+𝐴₂𝑉𝐵₂+𝐶₂𝑊𝐷₂+𝐴₃𝑉^𝐻𝐵₃+𝐶₃𝑊^𝐻𝐷₃+𝐴₄𝑉^𝑇𝐵₄+𝐶₄𝑊^𝑇𝐷₄=𝐸 (5) defined in (1) where 𝐴₁,𝐴₂,𝐶₁,𝐶₂∈ℂ^𝑚×𝑟, 𝐵₁,𝐵₂,𝐷₁,𝐷₂∈ℂ^𝑠×𝑛, 𝐴₃,𝐴₄,𝐶₃,𝐶₄∈ℂ^𝑚×𝑠, 𝐸∈ℂ^𝑚×𝑛 and 𝐵₃,𝐵₄,𝐷₃,𝐷₄∈ℂ^𝑟×𝑛 are given matrices, while 𝑉,𝑊∈ℂ^𝑟×𝑠 are matrices to be determined.

The following finite iterative algorithm is presented to solve it.

Algorithm 3. (1) Input 𝐴₁, 𝐴₂, 𝐶₁, 𝐶₂, 𝐵₁, 𝐵₂, 𝐷₁, 𝐷₂, 𝐴₃, 𝐴₄, 𝐶₃, 𝐶₄, 𝐵₃, 𝐵₄, 𝐷₃, 𝐷₄, 𝐸;
(2) Chosen arbitrary matrices 𝑉₁, 𝑊₁∈ℂ^𝑟×𝑠;
(3) Set 𝑅₁=𝐸−𝐴₁𝑉₁𝐵₁−𝐶₁𝑊₁𝐷₁−𝐴₂𝑉₁𝐵₂−𝐶₂𝑊₁𝐷₂−𝐴₃𝑉^𝐻₁𝐵₃−𝐶₃𝑊^𝐻₁𝐷₃−𝐴₄𝑉^𝑇₁𝐵₄−𝐶₄𝑊^𝑇₁𝐷₄,𝑃₁=𝐴^𝐻₁𝑅₁𝐵^𝐻₁+𝐴^𝐻₂𝑅₁𝐵^𝐻₂+𝐵₃𝑅^𝐻₁𝐴₃+𝐵₄𝑅^𝑇₁𝐴₄,𝑄₁=𝐶^𝐻₁𝑅₁𝐷^𝐻₁+𝐶^𝐻₂𝑅₁𝐷^𝐻₂+𝐷₃𝑅^𝐻₁𝐶₃+𝐷₄𝑅^𝑇₁𝐶₄,𝑘=∶1; (6)
(4) If 𝑅_𝑘=0, then stop; else go to Step 5;
(5) Set; 𝑉_𝑘+1=𝑉_𝑘+𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²𝑃_𝑘,𝑊_𝑘+1=𝑊_𝑘+𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²𝑄_𝑘,𝑅_𝑘+1=𝐸−𝐴₁𝑉_𝑘+1𝐵₁−𝐶₁𝑊_𝑘+1𝐷₁−𝐴₂𝑉_𝑘+1𝐵₂−𝐶₂𝑊_𝑘+1𝐷₂−𝐴₃𝑉^𝐻_𝑘+1𝐵₃−𝐶₃𝑊^𝐻_𝑘+1𝐷₃−𝐴₄𝑉^𝑇_𝑘+1𝐵₄−𝐶₄𝑊^𝑇_𝑘+1𝐷₄,𝑃_𝑘+1=𝐴^𝐻₁𝑅_𝑘+1𝐵^𝐻₁+𝐴^𝐻₂𝑅_𝑘+1𝐵^𝐻₂+𝐵₃𝑅^𝐻_𝑘+1𝐴₃+𝐵₄𝑅^𝑇_𝑘+1𝐴₄+𝑅_𝑘+1²𝑅_𝑘²𝑃_𝑘,𝑄_𝑘+1=𝐶^𝐻₁𝑅_𝑘+1𝐷^𝐻₁+𝐶^𝐻₂𝑅_𝑘+1𝐷^𝐻₂+𝐷₃𝑅^𝐻_𝑘+1𝐶₃+𝐷₄𝑅^𝑇_𝑘+1𝐶₄+𝑅_𝑘+1²𝑅_𝑘²𝑄_𝑘; (7)
(6) If 𝑅_𝑘+1=0, then stop; else let 𝑘=𝑘+1 go to Step 5.
To prove the convergence property of Algorithm 3, we first establish the following basic properties.

Lemma 4. Suppose that the matrix equation (1) is consistent and 𝑉^∗, 𝑊^∗ are arbitrary solutions of (1). Then for any initial matrices 𝑉₁ and 𝑊₁, we have Retr𝑃^𝐻_𝑖𝑉^∗−𝑉_𝑖+𝑄^𝐻_𝑖𝑊^∗−𝑊_𝑖=𝑅_𝑖², (8) where the sequence {𝑉_𝑖}, {𝑃_𝑖},  {𝑊_𝑖},  {𝑄_𝑖}, and {𝑅_𝑖} are generated by Algorithm 3 for 𝑖=1,2,….

Proof. We apply mathematical induction to prove the conclusion.
For 𝑖=1, from Algorithm 3 we have Retr𝑃^𝐻₁𝑉^∗−𝑉₁+𝑄^𝐻₁𝑊^∗−𝑊₁=Retr𝐴^𝐻₁𝑅₁𝐵^𝐻₁+𝐴^𝐻₂𝑅₁𝐵^𝐻₂+𝐵₃𝑅^𝐻₁𝐴₃+𝐵₄𝑅^𝑇₁𝐴₄𝐵^𝐻₂^𝐻𝑉^∗−𝑉₁+𝐶^𝐻₁𝑅₁𝐷^𝐻₁+𝐶^𝐻₂𝑅₁𝐷^𝐻₂+𝐷₃𝑅^𝐻₁𝐶₃+𝐷₄𝑅^𝑇₁𝐶₄𝐷^𝐻₂^𝐻×𝑊^∗−𝑊₁=Retr𝑅^𝐻₁𝐴₁𝑉^∗−𝑉₁𝐵₁+𝑅₁^𝐻𝐴₂𝑉^∗−𝑉₁𝐵₂+𝑅₁𝐵^𝐻₃𝑉^∗−𝑉₁𝐴^𝐻₃+𝑅₁𝐵^𝐻₄𝑉^∗−𝑉₁𝐴^𝐻₄+𝑅^𝐻₁𝐶₁𝑊^∗−𝑊₁𝐷₁+𝑅₁^𝐻𝐶₂𝑊^∗−𝑊₁𝐷₂+𝑅₁𝐷^𝐻₃𝑊^∗−𝑊₁𝐶^𝐻₃+𝑅₁𝐷^𝐻₄𝑊^∗−𝑊₁𝐶^𝐻₄. (9) From properties of trace and conjugate Retr𝑃^𝐻₁𝑉^∗−𝑉₁+𝑄^𝐻₁𝑊^∗−𝑊₁=Retr𝑅^𝐻₁𝐴₁𝑉^∗−𝑉₁𝐵₁+𝑅₁^𝐻𝐴₂𝑉^∗−𝑉₁𝐵₂+𝐴₃𝑉^∗−𝑉₁^𝐻𝐵₃𝑅^𝐻₁+𝐴₄𝑉^∗−𝑉₁^𝑇𝐵₄𝑅^𝐻₁+𝑅^𝐻₁𝐶₁𝑊^∗−𝑊₁𝐷₁+𝑅₁^𝐻𝐶₂𝑊^∗−𝑊₁𝐷₂+𝐶₃𝑊^∗−𝑊₁^𝐻𝐷₃𝑅^𝐻₁+𝐶₄𝑊^∗−𝑊₁^𝑇𝐷₄𝑅^𝐻₁=Retr𝑅^𝐻₁𝐴₁𝑉^∗−𝑉₁𝐵₁+𝑅^𝐻₁𝐴₂𝑉^∗−𝑉₁𝐵₂+𝑅^𝐻₁𝐴₃𝑉^∗−𝑉₁^𝐻𝐵₃+𝑅^𝐻₁𝐴₄𝑉^∗−𝑉₁^𝑇𝐵₄+𝑅^𝐻₁𝐶₁𝑊^∗−𝑊₁𝐷₁+𝑅^𝐻₁𝐶₂𝑊^∗−𝑊₁𝐷₂+𝑅^𝐻₁𝐶₃𝑊^∗−𝑊₁^𝐻𝐷₃+𝑅^𝐻₁𝐶₄𝑊^∗−𝑊₁^𝑇𝐷₄=Retr𝑅^𝐻₁𝐴₁𝑉^∗𝐵₁+𝐶₁𝑊^∗𝐷₁+𝐴₂𝑉^∗𝐵₂+𝐶₂𝑊^∗𝐷₂+𝐴₃𝑉^∗𝐻𝐵₃+𝐶₃𝑊^∗𝐻𝐷₃+𝐴₄𝑉^∗𝑇𝐵₄+𝐶₄𝑊^∗𝑇𝐷₄−𝐴₁𝑉₁𝐵₁−𝐶₁𝑊₁𝐷₁−𝐴₂𝑉₁𝐵₂−𝐶₂𝑊₁𝐷₂−𝐴₃𝑉^𝐻₁𝐵₃−𝐶₃𝑊^𝐻₁𝐷₃−𝐴₄𝑉^𝑇₁𝐵₄−𝐶₄𝑊^𝑇₁𝐷₄. (10) In view that 𝑉^∗ and 𝑊^∗ are solutions of matrix equation (1), with this relation we have Retr𝑃^𝐻₁𝑉^∗−𝑉₁+𝑄^𝐻₁𝑊^∗−𝑊₁=Retr𝑅^𝐻₁𝐸−𝐴₁𝑉₁𝐵₁−𝐶₁𝑊₁𝐷₁−𝐴₂𝑉₁𝐵₂−𝐶₂𝑊₁𝐷₂−𝐴₃𝑉^𝐻₁𝐵₃−𝐶₃𝑊^𝐻₁𝐷₃−𝐴₄𝑉^𝑇₁𝐵₄−𝐶₄𝑊^𝑇₁𝐷₄=Retr𝑅^𝐻₁𝑅₁=𝑅₁². (11) This implies that (8) holds for 𝑖=1.
Assume that (8) holds for =𝑘. That is, Retr𝑃^𝐻_𝑘𝑉^∗−𝑉_𝑘+𝑄^𝐻_𝑘𝑊^∗−𝑊_𝑘=𝑅_𝑘². (12) Then we have to prove that the conclusion holds for 𝑖=𝑘+1; it follows from Algorithm 3 that Retr𝑃^𝐻_𝑘+1𝑉^∗−𝑉_𝑘+1+𝑄^𝐻_𝑘+1𝑊^∗−𝑊_𝑘+1=Retr𝐴^𝐻₁𝑅_𝑘+1𝐵^𝐻₁+𝐴^𝐻₂𝑅_𝑘+1𝐵^𝐻₂+𝐵₃𝑅^𝐻_𝑘+1𝐴₃+𝐵₄𝑅^𝑇_𝑘+1𝐴₄+𝑅_𝑘+1²𝑅_𝑘²𝑃_𝑘^𝐻𝑉^∗−𝑉_𝑘+1+𝐶^𝐻₁𝑅_𝑘+1𝐷^𝐻₁+𝐶^𝐻₂𝑅_𝑘+1𝐷^𝐻₂+𝐷₃𝑅^𝐻_𝑘+1𝐶₃+𝐷₄𝑅^𝑇_𝑘+1𝐶₄+𝑅_𝑘+1²𝑅_𝑘²𝑄_𝑘^𝐻𝑊^∗−𝑊_𝑘+1=Retr𝑅^𝐻_𝑘+1𝐴₁𝑉^∗−𝑉_𝑘+1𝐵₁+𝑅^𝑇_𝑘+1𝐴₂𝑉^∗−𝑉_𝑘+1𝐵₂+𝑅_𝑘+1𝐵^𝐻₃𝑉^∗−𝑉_𝑘+1𝐴^𝐻₃+𝑅_𝑘+1𝐵₄^𝐻𝑉^∗−𝑉_𝑘+1𝐴₄^𝐻+𝑅^𝐻_𝑘+1𝐶₁𝑊^∗−𝑊_𝑘+1𝐷₁+𝑅^𝑇_𝑘+1𝐶₂𝑊^∗−𝑊_𝑘+1𝐷₂+𝑅_𝑘+1𝐷^𝐻₃𝑊^∗−𝑊_𝑘+1𝐶^𝐻₃+𝑅_𝑘+1𝐷₄^𝐻𝑊^∗−𝑊_𝑘+1𝐶₄^𝐻+𝑅_𝑘+1²𝑅_𝑘²𝑃^𝐻_𝑘𝑉^∗−𝑉_𝑘+1+𝑄^𝐻_𝑘𝑊^∗−𝑊_𝑘+1. (13) From properties of trace and conjugate we get Retr𝑃^𝐻_𝑘+1𝑉^∗−𝑉_𝑘+1+𝑄^𝐻_𝑘+1𝑊^∗−𝑊_𝑘+1=Retr𝑅^𝐻_𝑘+1𝐴₁𝑉^∗−𝑉_𝑘+1𝐵₁+𝑅^𝑇_𝑘+1𝐴₂𝑉^∗−𝑉_𝑘+1𝐵₂+𝐴₃𝑉^∗−𝑉_𝑘+1^𝐻𝐵₃𝑅^𝐻_𝑘+1+𝐴₄𝑉^∗−𝑉_𝑘+1^𝑇𝐵₄𝑅^𝐻_𝑘+1+𝑅^𝐻_𝑘+1𝐶₁𝑊^∗−𝑊_𝑘+1𝐷₁+𝑅^𝑇_𝑘+1𝐶₂𝑊^∗−𝑊_𝑘+1𝐷₂+𝐶₃𝑊^∗−𝑊_𝑘+1^𝐻𝐷₃𝑅^𝐻_𝑘+1+𝐶₄𝑊^∗−𝑊_𝑘+1^𝑇𝐷₄𝑅^𝐻_𝑘+1+𝑅_𝑘+1²𝑅_𝑘²×𝑃^𝐻_𝑘𝑉^∗−𝑉_𝑘+1+𝑄^𝐻_𝑘𝑊^∗−𝑊_𝑘+1=Retr𝑅^𝐻_𝑘+1𝐴₁𝑉^∗−𝑉_𝑘+1𝐵₁+𝑅^𝐻_𝑘+1𝐴₂𝑉^∗−𝑉_𝑘+1𝐵₂+𝑅^𝐻_𝑘+1𝐴₃𝑉^∗−𝑉_𝑘+1^𝐻𝐵₃+𝑅^𝐻_𝑘+1𝐴₄𝑉^∗−𝑉_𝑘+1^𝑇𝐵₄+𝑅^𝐻_𝑘+1𝐶₁𝑊^∗−𝑊_𝑘+1𝐷₁+𝑅^𝐻_𝑘+1𝐶₂𝑊^∗−𝑊_𝑘+1𝐷₂+𝑅^𝐻_𝑘+1𝐶₃𝑊^∗−𝑊_𝑘+1^𝐻𝐷₃+𝑅^𝐻_𝑘+1𝐶₄𝑊^∗−𝑊_𝑘+1^𝑇𝐷₄+𝑅_𝑘+1²𝑅_𝑘²×Retr𝑃^𝐻_𝑘𝑉^∗−𝑉_𝑘−𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²𝑃_𝑘+𝑄^𝐻_𝑘𝑊^∗−𝑊_𝑘−𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²𝑄_𝑘=Retr𝑅^𝐻_𝑘+1𝐴₁𝑉^∗𝐵₁+𝐶₁𝑊^∗𝐷₁+𝐴₂𝑉^∗𝐵₂+𝐶₂𝑊^∗𝐷₂+𝐴₃𝑉^∗𝐻𝐵₃+𝐶₃𝑊^∗𝐻𝐷₃+𝐴₄𝑉^∗𝑇𝐵₄+𝐶₄𝑊^∗𝑇𝐷₄−𝐴₁𝑉_𝑘+1𝐵₁−𝐶₁𝑊_𝑘+1𝐷₁−𝐴₂𝑉_𝑘+1𝐵₂−𝐶₂𝑊_𝑘+1𝐷₂−𝐴₃𝑉^𝐻_𝑘+1𝐵₃−𝐶₃𝑊^𝐻_𝑘+1𝐷₃−𝐴₄𝑉^𝑇_𝑘+1𝐵₄−𝐶₄𝑊^𝑇_𝑘+1𝐷₄+𝑅_𝑘+1²𝑅_𝑘²Retr𝑃^𝐻_𝑘𝑉^∗−𝑉_𝑘+𝑄^𝐻_𝑘𝑊^∗−𝑊_𝑘−𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²𝑃^𝐻_𝑘𝑃_𝑘+𝑄^𝐻_𝑘𝑄_𝑘. (14) In view that 𝑉^∗ and 𝑊^∗ are solutions of matrix equation (1), with relation (14) one has Retr𝑃^𝐻_𝑘+1𝑉^∗−𝑉_𝑘+1+𝑄^𝐻_𝑘+1𝑊^∗−𝑊_𝑘+1=Retr𝑅^𝐻_𝑘+1𝐸−𝐴₁𝑉_𝑘+1𝐵₁−𝐶₁𝑊_𝑘+1𝐷₁−𝐴₂𝑉_𝑘+1𝐵₂−𝐶₂𝑊_𝑘+1𝐷₂−𝐴₃𝑉^𝐻_𝑘+1𝐵₃−𝐶₃𝑊^𝐻_𝑘+1𝐷₃−𝐴₄𝑉^𝑇_𝑘+1𝐵₄−𝐶₄𝑊^𝑇_𝑘+1𝐷₄+𝑅_𝑘+1²𝑅_𝑘²𝑅_𝑘²−𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²𝑃_𝑘²+𝑄_𝑘²=Retr𝑅^𝐻_𝑘+1𝑅_𝑘+1=𝑅_𝑘+1². (15) Then relation (8) holds by mathematical induction.

Lemma 5. Suppose that the matrix equation (1) is consistent and the sequences {𝑅_𝑖}, {𝑃_𝑖}, and {𝑄_𝑖} are generated by Algorithm 3 with any initial matrices 𝑉₁, 𝑊₁, such that 𝑅_𝑖≠0for all 𝑖=1,2,…,𝑘, and then Retr𝑅^𝐻_𝑗𝑅_𝑖𝑃=0,Retr^𝐻_𝑗𝑃_𝑖+𝑄^𝐻_𝑗𝑄_𝑖=0,𝑖,𝑗=1,2,…,𝑘,𝑖≠𝑗. (16)

Proof. We apply mathematical induction.
Step 1. We prove that Retr𝑅^𝐻_𝑖+1𝑅_𝑖𝑃=0,Retr^𝐻_𝑖+1𝑃_𝑖+𝑄^𝐻_𝑖+1𝑄_𝑖=0,𝑖=1,2,…,𝑘. (17)(18) First from Algorithm 3 we have 𝑅_𝑘+1=𝐸−𝐴₁𝑉_𝑘+1𝐵₁−𝐶₁𝑊_𝑘+1𝐷₁−𝐴₂𝑉_𝑘+1𝐵₂−𝐶₂𝑊_𝑘+1𝐷₂−𝐴₃𝑉^𝐻_𝑘+1𝐵₃−𝐶₃𝑊^𝐻_𝑘+1𝐷₃−𝐴₄𝑉^𝑇_𝑘+1𝐵₄−𝐶₄𝑊^𝑇_𝑘+1𝐷₄,𝑅_𝑘+1=𝐸−𝐴₁𝑉_𝑘+𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²𝑃_𝑘𝐵₁−𝐶₁𝑊_𝑘+𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²𝑄_𝑘𝐷₁−𝐴₂𝑉_𝑘+𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²𝑃_𝑘𝐵₂−𝐶₂𝑊_𝑘+𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²𝑄_𝑘𝐷₂−𝐴₃𝑉_𝑘+𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²𝑃_𝑘^𝐻𝐵₃−𝐶₃𝑊_𝑘+𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²𝑄_𝑘^𝐻𝐷₃−𝐴₄𝑉_𝑘+𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²𝑃_𝑘^𝑇𝐵₄−𝐶₄𝑊_𝑘+𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²𝑄_𝑘^𝑇𝐷₄=𝐸−𝐴₁𝑉_𝑘𝐵₁−𝐶₁𝑊_𝑘𝐷₁−𝐴₂𝑉_𝑘𝐵₂−𝐶₂𝑊_𝑘𝐷₂−𝐴₃𝑉^𝐻_𝑘𝐵₃−𝐶₃𝑊^𝐻_𝑘𝐷₃−𝐴₄𝑉^𝑇_𝑘𝐵₄−𝐶₄𝑊^𝑇_𝑘𝐷₄−𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²×𝐴₁𝑃_𝑘𝐵₁+𝐶₁𝑄_𝑘𝐷₁+𝐴₂𝑃_𝑘𝐵₂+𝐶₂𝑄_𝑘𝐷₂+𝐴₃𝑃^𝐻_𝑘𝐵₃+𝐶₃𝑄^𝐻_𝑘𝐷₃+𝐴₄𝑃^𝑇_𝑘𝐵₄+𝐶₄𝑄^𝑇_𝑘𝐷₄,𝑅_𝑘+1=𝑅_𝑘−𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²×𝐴₁𝑃_𝑘𝐵₁+𝐶₁𝑄_𝑘𝐷₁+𝐴₂𝑃_𝑘𝐵₂+𝐶₂𝑄_𝑘𝐷₂+𝐴₃𝑃^𝐻_𝑘𝐵₃+𝐶₃𝑄^𝐻_𝑘𝐷₃+𝐴₄𝑃^𝑇_𝑘𝐵₄+𝐶₄𝑄^𝑇_𝑘𝐷₄. (19) For 𝑖=1, it follows from (19) that Retr𝑅^𝐻₂𝑅₁=Retr𝑅₁−𝑅₁²𝑃₁²+𝑄₁²×𝐴₁𝑃₁𝐵₁+𝐶₁𝑄₁𝐷₁+𝐴₂𝑃₁𝐵₂+𝐶₂𝑄₁𝐷₂+𝐴₃𝑃^𝐻₁𝐵₃+𝐶₃𝑄^𝐻₁𝐷₃+𝐴₄𝑃^𝑇₁𝐵₄+𝐶₄𝑄^𝑇₁𝐷₄^𝐻𝑅₁=Retr𝑅^𝐻₁𝑅₁−𝑅₁²𝑃₁²+𝑄₁²×tr𝑃^𝐻₁𝐴^𝐻₁𝑅₁𝐵^𝐻₁+𝑄^𝐻₁𝐶^𝐻₁𝑅₁𝐷^𝐻₁+𝑃₁^𝐻𝐴^𝐻₂𝑅₁𝐵^𝐻₂+𝑄₁^𝐻𝐶^𝐻₂𝑅₁𝐷^𝐻₂+𝑃₁𝐴^𝐻₃𝑅₁𝐵^𝐻₃+𝑄₁𝐶^𝐻₃𝑅₁𝐷^𝐻₃+𝑃₁𝐴^𝐻₄𝑅₁𝐵^𝐻₄+𝑄₁𝐶^𝐻₄𝑅₁𝐷^𝐻₄=𝑅₁²−𝑅₁²𝑃₁²+𝑄₁²×Retr𝑃^𝐻₁𝐴^𝐻₁𝑅₁𝐵^𝐻₁+𝑄^𝐻₁𝐶^𝐻₁𝑅₁𝐷^𝐻₁+𝑃₁^𝐻𝐴^𝐻₂𝑅₁𝐵^𝐻₂+𝑄₁^𝐻𝐶^𝐻₂𝑅₁𝐷^𝐻₂+𝐵₃𝑅^𝐻₁𝐴₃𝑃^𝐻₁+𝐷₃𝑅^𝐻₁𝐶₃𝑄^𝐻₁+𝐵₄𝑅^𝑇₁𝐴₄𝑃^𝐻₁+𝐷₄𝑅^𝑇₁𝐶₄𝑄^𝐻₁=𝑅₁²−𝑅₁²𝑃₁²+𝑄₁²×Retr𝑃^𝐻₁𝐴^𝐻₁𝑅₁𝐵^𝐻₁+𝐴^𝐻₂𝑅₁𝐵^𝐻₂+𝐵₃𝑅^𝐻₁𝐴₃+𝐵₄𝑅^𝑇₁𝐴₄+𝑄^𝐻₁𝐶^𝐻₁𝑅₁𝐷^𝐻₁+𝐶^𝐻₂𝑅₁𝐷^𝐻₂+𝐷₃𝑅^𝐻₁𝐶₃+𝐷₄𝑅^𝑇₁𝐶₄=𝑅₁²−𝑅₁²𝑃₁²+𝑄₁²Retr𝑃^𝐻₁𝑃₁+𝑄^𝐻₁𝑄₁,Retr𝑅^𝐻₂𝑅₁=𝑅₁²−𝑅₁²𝑃₁²+𝑄₁²𝑃₁²+𝑄₁²=0. (20) This implies that (17) is satisfied for 𝑖=1.
From Algorithm 3 we have Retr𝑃^𝐻₂𝑃₁+𝑄^𝐻₂𝑄₁=Retr𝐴^𝐻₁𝑅₂𝐵^𝐻₁+𝐴^𝐻₂𝑅₂𝐵^𝐻₂+𝐵₃𝑅^𝐻₂𝐴₃+𝐵₄𝑅^𝑇₂𝐴₄+𝑅₂²𝑅₁²𝑃₁^𝐻𝑃₁+𝐶^𝐻₁𝑅₂𝐷^𝐻₁+𝐶^𝐻₂𝑅₂𝐷^𝐻₂+𝐷₃𝑅^𝐻₂𝐶₃+𝐷₄𝑅^𝑇₂𝐶₄+𝑅₂²𝑅₁²𝑄₁^𝐻𝑄₁=Retr𝑅^𝐻₂𝐴₁𝑃₁𝐵₁+𝑅₂^𝐻𝐴₂𝑃₁𝐵₂+𝑅₂𝐵^𝐻₃𝑃₁𝐴^𝐻₃+𝑅₂𝐵₄^𝐻𝑃₁𝐴₄^𝐻+𝑅₂²𝑅₁²𝑃^𝐻₁𝑃₁+𝑅^𝐻₂𝐶₁𝑄₁𝐷₁+𝑅₂^𝐻𝐶₂𝑄₁𝐷₂+𝑅₂𝐷^𝐻₃𝑄₁𝐶^𝐻₃+𝑅₂𝐷₄^𝐻𝑄₁𝐶₄^𝐻+𝑅₂²𝑅₁²𝑄^𝐻₁𝑄₁=Retr𝑅^𝐻₂𝐴₁𝑃₁𝐵₁+𝑅₂^𝐻𝐴₂𝑃₁𝐵₂+𝐴₃𝑃^𝐻₁𝐵₃𝑅^𝐻₂+𝐴₄𝑃^𝑇₁𝐵₄𝑅^𝐻₂+𝑅^𝐻₂𝐶₁𝑄₁𝐷₁+𝑅₂^𝐻𝐶₂𝑄₁𝐷₂+𝐶₃𝑄^𝐻₁𝐷₃𝑅^𝐻₂+𝐶₄𝑄^𝑇₁𝐷₄𝑅^𝐻₂+𝑅₂²𝑅₁²𝑃^𝐻₁𝑃₁+𝑄^𝐻₁𝑄₁=Retr𝑅^𝐻₂𝐴₁𝑃₁𝐵₁+𝑅^𝐻₂𝐴₂𝑃₁𝐵₂+𝑅^𝐻₂𝐴₃𝑃^𝐻₁𝐵₃+𝑅^𝐻₂𝐴₄𝑃^𝑇₁𝐵₄+𝑅^𝐻₂𝐶₁𝑄₁𝐷₁+𝑅^𝐻₂𝐶₂𝑄₁𝐷₂+𝑅^𝐻₂𝐶₃𝑄^𝐻₁𝐷₃+𝑅^𝐻₂𝐶₄𝑄^𝑇₁𝐷₄+𝑅₂²𝑅₁²𝑃^𝐻₁𝑃₁+𝑄^𝐻₁𝑄₁=Retr𝑅^𝐻₂𝐴₁𝑃₁𝐵₁+𝐶₁𝑄₁𝐷₁+𝐴₂𝑃₁𝐵₂+𝐶₂𝑄₁𝐷₂+𝐴₃𝑃^𝐻₁𝐵₃+𝐶₃𝑄^𝐻₁𝐷₃+𝐴₄𝑃^𝑇₁𝐵₄+𝐶₄𝑄^𝐻₁𝐷₄+𝑅₂²𝑅₁²𝑃₁²+𝑄₁²=𝑃₁²+𝑄₁²𝑅₁²Retr𝑅^𝐻₂𝑅₁−𝑅₂+𝑅₂²𝑅₁²𝑃₁²+𝑄₁²𝑃=−¹²+𝑄₁²𝑅₁²𝑅₂²+𝑅₂²𝑅₁²𝑃₁²+𝑄₁²=0. (21) This implies that (18) is satisfied for 𝑖 = 1.
Assume that (17) and (18) hold for 𝑖=𝑘−1, from Algorithm 3 we have Retr𝑅^𝐻_𝑘+1𝑅_𝑘=Retr𝑅_𝑘−𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²×𝐴₁𝑃_𝑘𝐵₁+𝐶₁𝑄_𝑘𝐷₁+𝐴₂𝑃_𝑘𝐵₂+𝐶₂𝑄_𝑘𝐷₂+𝐴₃𝑃^𝐻_𝑘𝐵₃+𝐶₃𝑄^𝐻_𝑘𝐷₃+𝐴₄𝑃^𝑇_𝑘𝐵₄+𝐶₄𝑄^𝑇_𝑘𝐷₄^𝐻𝑅_𝑘,Retr𝑅^𝐻_𝑘+1𝑅_𝑘=Retr𝑅^𝐻_𝑘𝑅_𝑘−𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²×tr𝑃^𝐻_𝑘𝐴^𝐻₁𝑅_𝑘𝐵^𝐻₁+𝑄^𝐻_𝑘𝐶^𝐻₁𝑅_𝑘𝐷^𝐻₁+𝑃_𝑘^𝐻𝐴^𝐻₂𝑅_𝑘𝐵^𝐻₂+𝑄_𝑘^𝐻𝐶^𝐻₂𝑅_𝑘𝐷^𝐻₂+𝑃_𝑘𝐴^𝐻₃𝑅_𝑘𝐵^𝐻₃+𝑄_𝑘𝐶^𝐻₃𝑅_𝑘𝐷^𝐻₃+𝑃_𝑘𝐴^𝐻₄𝑅_𝑘𝐵^𝐻₄+𝑄_𝑘𝐶^𝐻₄𝑅_𝑘𝐷^𝐻₄=𝑅_𝑘²−𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²×Retr𝑃^𝐻_𝑘𝐴^𝐻₁𝑅_𝑘𝐵^𝐻₁+𝑄^𝐻_𝑘𝐶^𝐻₁𝑅_𝑘𝐷^𝐻₁+𝑃_𝑘^𝐻𝐴^𝐻₂𝑅_𝑘𝐵^𝐻₂+𝑄_𝑘^𝐻𝐶^𝐻₂𝑅_𝑘𝐷^𝐻₂+𝐵₃𝑅^𝐻_𝑘𝐴₃𝑃^𝐻_𝑘+𝐷₃𝑅^𝐻_𝑘𝐶₃𝑄^𝐻_𝑘+𝐵₄𝑅^𝑇_𝑘𝐴₄𝑃^𝐻_𝑘+𝐷₄𝑅^𝑇_𝑘𝐶₄𝑄^𝐻_𝑘=𝑅₁²−𝑅₁²𝑃_𝑘²+𝑄_𝑘²×Retr𝑃^𝐻_𝑘𝐴^𝐻₁𝑅_𝑘𝐵^𝐻₁+𝐴^𝐻₂𝑅_𝑘𝐵^𝐻₂+𝐵₃𝑅^𝐻_𝑘𝐴₃+𝐵₄𝑅^𝑇_𝑘𝐴₄+𝑄^𝐻_𝑘𝐶^𝐻₁𝑅_𝑘𝐷^𝐻₁+𝐶^𝐻₂𝑅_𝑘𝐷^𝐻₂+𝐷₃𝑅^𝐻_𝑘𝐶₃+𝐷₄𝑅^𝑇_𝑘𝐶₄=𝑅_𝑘²−𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²×Retr𝑃^𝐻_𝑘𝑃_𝑘−𝑅_𝑘²𝑅_𝑘−1²𝑃_𝑘−1+𝑄^𝐻_𝑘𝑄_𝑘−𝑅_𝑘²𝑅_𝑘−1²𝑄_𝑘−1=𝑅_𝑘²−𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²𝑃_𝑘²+𝑄_𝑘²=0. (22) Thus (17) holds for 𝑖=𝑘.
Also, from Algorithm 3 we have Retr𝑃^𝐻_𝑘+1𝑃_𝑘+𝑄^𝐻_𝑘+1𝑄_𝑘=Retr𝐴^𝐻₁𝑅_𝑘+1𝐵^𝐻₁+𝐴^𝐻₂𝑅_𝑘+1𝐵^𝐻₂+𝐵₃𝑅^𝐻_𝑘+1𝐴₃+𝐵₄𝑅^𝑇_𝑘+1𝐴₄+𝑅_𝑘+1²𝑅_𝑘²𝑃_𝑘^𝐻𝑃_𝑘+𝐶^𝐻₁𝑅_𝑘+1𝐷^𝐻₁+𝐶^𝐻₂𝑅_𝑘+1𝐷^𝐻₂+𝐷₃𝑅^𝐻_𝑘+1𝐶₃+𝐷₄𝑅^𝑇_𝑘+1𝐶₄+𝑅_𝑘+1²𝑅_𝑘²𝑄_𝑘^𝐻𝑄_𝑘=Retr𝑅^𝐻_𝑘+1𝐴₁𝑃_𝑘𝐵₁+𝑅^𝐻_𝑘+1𝐴₂𝑃_𝑘𝐵₂+𝑅_𝑘+1𝐵^𝐻₃𝑃_𝑘𝐴^𝐻₃+𝑅_𝑘+1𝐵₄^𝐻𝑃_𝑘𝐴₄^𝐻+𝑅_𝑘+1²𝑅_𝑘²𝑃^𝐻_𝑘𝑃_𝑘+𝑅^𝐻_𝑘+1𝐶₁𝑄_𝑘𝐷₁+𝑅^𝐻_𝑘+1𝐶₂𝑄_𝑘𝐷₂+𝑅_𝑘+1𝐷^𝐻₃𝑄_𝑘𝐶^𝐻₃+𝑅_𝑘+1𝐷₄^𝐻𝑄_𝑘𝐶₄^𝐻+𝑅_𝑘+1²𝑅_𝑘²𝑄^𝐻_𝑘𝑄_𝑘=Retr𝑅^𝐻_𝑘+1𝐴₁𝑃_𝑘𝐵₁+𝑅^𝐻_𝑘+1𝐴₂𝑃_𝑘𝐵₂+𝐴₃𝑃^𝐻_𝑘𝐵₃𝑅^𝐻_𝑘+1+𝐴₄𝑃^𝑇_𝑘𝐵₄𝑅^𝐻_𝑘+1+𝑅_𝑘+1²𝑅_𝑘²𝑃^𝐻_𝑘𝑃_𝑘+𝑅^𝐻_𝑘+1𝐶₁𝑄_𝑘𝐷₁+𝑅^𝐻_𝑘+1𝐶₂𝑄_𝑘𝐷₂+𝐶₃𝑄^𝐻_𝑘𝐷₃𝑅^𝐻_𝑘+1+𝐶₄𝑄^𝑇_𝑘𝐷₄𝑅^𝐻_𝑘+1+𝑅_𝑘+1²𝑅_𝑘²𝑄^𝐻_𝑘𝑄_𝑘=Retr𝑅^𝐻_𝑘+1𝐴₁𝑃_𝑘𝐵₁+𝐶₁𝑄_𝑘𝐷₁+𝐴₂𝑃_𝑘𝐵₂+𝐶₂𝑄_𝑘𝐷₂+𝐴₃𝑃^𝐻_𝑘𝐵₃+𝐶₃𝑄^𝐻_𝑘𝐷₃+𝐴₄𝑃^𝑇_𝑘𝐵₄+𝐶₄𝑄^𝑇_𝑘𝐷₄+𝑅_𝑘+1²𝑅_𝑘²𝑃^𝐻_𝑘𝑃_𝑘+𝑄^𝐻_𝑘𝑄_𝑘=𝑃_𝑘²+𝑄_𝑘²𝑅_𝑘²Retr𝑅^𝐻_𝑘+1𝑅_𝑘−𝑅_𝑘+1+𝑅_𝑘+1²𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²𝑃=−^𝑘2+𝑄_𝑘²𝑅_𝑘²𝑅_𝑘+1²+𝑅_𝑘+1²𝑅_𝑘²𝑃_𝑘²+𝑄_𝑘²=0. (23) This implies that (17) and (18) hold for 𝑖=𝑘.
Then relations (17) and (18) holds by mathematical induction.
Step 2. We want to show that Retr𝑅^𝐻_𝑖+𝑙𝑅_𝑖𝑃=0,Retr^𝐻_𝑖+𝑙𝑃_𝑖+𝑄^𝐻_𝑖+𝑙𝑄_𝑖=0 (24) holds for 𝑙≥1. We will prove this conclusion by induction. The case of 𝑙=1 has been proven in Step 1. Now we assume that (24) holds for 𝑙≤𝑠, 𝑠≥1. The aim is to show that Retr𝑅^𝐻_𝑖+𝑠+1𝑅_𝑖𝑃=0,Retr^𝐻_𝑖+𝑠+1𝑃_𝑖+𝑄^𝐻_𝑖+𝑠+1𝑄_𝑖=0. (25) First we prove the following: Retr𝑅^𝐻_𝑠+1𝑅₀𝑃=0,Retr^𝐻_𝑠+1𝑃₀+𝑄^𝐻_𝑠+1𝑄₀=0. (26) By using Algorithm 3, from (19) and induction we have Retr𝑅^𝐻_𝑠+1𝑅₀=Retr𝑅_𝑠−𝑅_𝑠²𝑃_𝑠²+𝑄_𝑠²×𝐴₁𝑃_𝑠𝐵₁+𝐶₁𝑄_𝑠𝐷₁+𝐴₂𝑃_𝑠𝐵₂+𝐶₂𝑄_𝑠𝐷₂+𝐴₃𝑃^𝐻_𝑠𝐵₃+𝐶₃𝑄^𝐻_𝑠𝐷₃+𝐴₄𝑃^𝑇_𝑠𝐵₄+𝐶₄𝑄^𝑇_𝑠𝐷₄^𝐻𝑅₀=Retr𝑅^𝐻_𝑠𝑅₀−𝑅_𝑠²𝑃_𝑠²+𝑄_𝑠²×tr𝑃^𝐻_𝑠𝐴^𝐻₁𝑅₀𝐵^𝐻₁+𝑄^𝐻_𝑠𝐶^𝐻₁𝑅₀𝐷^𝐻₁+𝑃_𝑠^𝐻𝐴^𝐻₂𝑅₀𝐵^𝐻₂+𝑄_𝑠^𝐻𝐶^𝐻₂𝑅₀𝐷^𝐻₂+𝑃_𝑠𝐴^𝐻₃𝑅₀𝐵^𝐻₃+𝑄_𝑠𝐶^𝐻₃𝑅₀𝐷^𝐻₃+𝑃_𝑠𝐴^𝐻₄𝑅₀𝐵^𝐻₄+𝑄_𝑠𝐶^𝐻₄𝑅₀𝐷^𝐻₄,Retr𝑅^𝐻_𝑠+1𝑅₀=Retr𝑅^𝐻_𝑠𝑅₀−𝑅_𝑠²𝑃_𝑠²+𝑄_𝑠²×Retr𝑃^𝐻_𝑠𝐴^𝐻₁𝑅₀𝐵^𝐻₁+𝑄^𝐻_𝑠𝐶^𝐻₁𝑅₀𝐷^𝐻₁+𝑃_𝑠^𝐻𝐴^𝐻₂𝑅₀𝐵^𝐻₂+𝑄_𝑠^𝐻𝐶^𝐻₂𝑅₀𝐷^𝐻₂+𝐵₃𝑅^𝐻₀𝐴₃𝑃^𝐻_𝑠+𝐷₃𝑅^𝐻₀𝐶₃𝑄^𝐻_𝑠+𝐵₄𝑅^𝑇₀𝐴₄𝑃^𝐻_𝑠+𝐷₄𝑅^𝑇₀𝐶₄𝑄^𝐻_𝑠𝑅=−^𝑠2𝑃_𝑠²+𝑄_𝑠²×Retr𝑃^𝐻_𝑠𝐴^𝐻₁𝑅₀𝐵^𝐻₁+𝐴^𝐻₂𝑅₀𝐵^𝐻₂+𝐵₃𝑅^𝐻₀𝐴₃+𝐵₄𝑅^𝑇₀𝐴₄+𝑄^𝐻_𝑠𝐶^𝐻₁𝑅₀𝐷^𝐻₁+𝐶^𝐻₂𝑅₀𝐷^𝐻₂+𝐷₃𝑅^𝐻₀𝐶₃+𝐷₄𝑅^𝑇₀𝐶₄𝑅=−^𝑠2𝑃_𝑠²+𝑄_𝑠²Retr𝑃^𝐻_𝑠𝑃₀+𝑄^𝐻_𝑠𝑄₀𝑃=0,Retr^𝐻_𝑠+1𝑃₀+𝑄^𝐻_𝑠+1𝑄₀=Retr𝐴^𝐻₁𝑅_𝑠+1𝐵^𝐻₁+𝐴^𝐻₂𝑅_𝑠+1𝐵^𝐻₂+𝐵₃𝑅^𝐻_𝑠+1𝐴₃+𝐵₄𝑅^𝑇_𝑠+1𝐴₄+𝑅_𝑠+1²𝑅_𝑠²𝑃_𝑠^𝐻𝑃₀+𝐶^𝐻₁𝑅_𝑠+1𝐷^𝐻₁+𝐶^𝐻₂𝑅_𝑠+1𝐷^𝐻₂+𝐷₃𝑅^𝐻_𝑠+1𝐶₃+𝐷₄𝑅^𝑇_𝑘+1𝐶₄+𝑅_𝑠+1²𝑅_𝑠²𝑄_𝑠^𝐻𝑄₀=Retr𝑅^𝐻_𝑠+1𝐴₁𝑃₀𝐵₁+𝑅^𝐻_𝑠+1𝐴₂𝑃₀𝐵₂+𝑅_𝑠+1𝐵^𝐻₃𝑃₀𝐴^𝐻₃+𝑅_𝑠+1𝐵₄^𝐻𝑃₀𝐴₄^𝐻+𝑅_𝑠+1²𝑅_𝑠²𝑃^𝐻_𝑠𝑃₀+𝑅^𝐻_𝑠+1𝐶₁𝑄₀𝐷₁+𝑅^𝐻_𝑠+1𝐶₂𝑄₀𝐷₂+𝑅_𝑠+1𝐷^𝐻₃𝑄₀𝐶^𝐻₃+𝑅_𝑠+1𝐷₄^𝐻𝑄₀𝐶₄^𝐻+𝑅_𝑠+1²𝑅_𝑠²𝑄^𝐻_𝑠𝑄₀=Retr𝑅^𝐻_𝑠+1𝐴₁𝑃₀𝐵₁+𝑅^𝐻_𝑠+1𝐴₂𝑃₀𝐵₂+𝐴₃𝑃^𝐻₀𝐵₃𝑅^𝐻_𝑠+1+𝐴₄𝑃^𝑇₀𝐵₄𝑅^𝐻_𝑠+1+𝑅_𝑠+1²𝑅_𝑠²𝑃^𝐻_𝑠𝑃₀+𝑅^𝐻_𝑠+1𝐶₁𝑄₀𝐷₁+𝑅^𝐻_𝑠+1𝐶₂𝑄₀𝐷₂+𝐶₃𝑄^𝐻₀𝐷₃𝑅^𝐻_𝑠+1+𝐶₄𝑄^𝑇₀𝐷₄𝑅^𝐻_𝑠+1+𝑅_𝑠+1²𝑅_𝑠²𝑄^𝐻_𝑠𝑄₀=Retr𝑅^𝐻_𝑠+1𝐴₁𝑃₀𝐵₁+𝐶₁𝑄₀𝐷₁+𝐴₂𝑃₀𝐵₂+𝐶₂𝑄₀𝐷₂+𝐴₃𝑃^𝐻₀𝐵₃+𝐶₃𝑄^𝐻₀𝐷₃+𝐴₄𝑃^𝑇₀𝐵₄+𝐶₄𝑄^𝑇₀𝐷₄+𝑅_𝑠+1²𝑅_𝑠²𝑃^𝐻_𝑠𝑃₀+𝑄^𝐻_𝑠𝑄₀=𝑃₀²+𝑄₀²𝑅₀²Retr𝑅^𝐻_𝑠+1𝑅₀−𝑅₁=0. (27) Then (26) is holds.
From Algorithm 3 we have Retr𝑃^𝐻_𝑖+𝑠+1𝑃_𝑖+𝑄^𝐻_𝑖+𝑠+1𝑄_𝑖=Retr𝐴^𝐻₁𝑅_𝑖+𝑠+1𝐵^𝐻₁+𝐴^𝐻₂𝑅_𝑖+𝑠+1𝐵^𝐻₂+𝐵₃𝑅^𝐻_𝑖+𝑠+1𝐴₃+𝐵₄𝑅^𝑇_𝑖+𝑠+1𝐴₄+𝑅_𝑖+𝑠+1²𝑅_𝑖+𝑠²𝑃_𝑖+𝑠^𝐻𝑃_𝑖+𝐶^𝐻₁𝑅_𝑖+𝑠+1𝐷^𝐻₁+𝐶^𝐻₂𝑅_𝑖+𝑠+1𝐷^𝐻₂+𝐷₃𝑅^𝐻_𝑖+𝑠+1𝐶₃+𝐷₄𝑅^𝑇_𝑖+𝑠+1𝐶₄+𝑅_𝑖+𝑠+1²𝑅_𝑖+𝑠²𝑄_𝑖+𝑠^𝐻𝑄_𝑖=Retr𝑅^𝐻_𝑖+𝑠+1𝐴₁𝑃_𝑖𝐵₁+𝑅^𝐻_𝑖+𝑠+1𝐴₂𝑃_𝑖𝐵₂+𝑅_𝑖+𝑠+1𝐵^𝐻₃𝑃_𝑖𝐴^𝐻₃+𝑅_𝑖+𝑠+1𝐵₄^𝐻𝑃_𝑖𝐴₄^𝐻+𝑅_𝑖+𝑠+1²𝑅_𝑖+𝑠²𝑃^𝐻_𝑖+𝑠𝑃_𝑖+𝑅^𝐻_𝑖+𝑠+1𝐶₁𝑄_𝑖𝐷₁+𝑅^𝐻_𝑖+𝑠+1𝐶₂𝑄_𝑖𝐷₂+𝑅_𝑖+𝑠+1𝐷^𝐻₃𝑄_𝑖𝐶^𝐻₃+𝑅_𝑖+𝑠+1𝐷₄^𝐻𝑄_𝑖𝐶₄^𝐻+𝑅_𝑖+𝑠+1²𝑅_𝑖+𝑠²𝑄^𝐻_𝑖+𝑠𝑄_𝑖=Retr𝑅^𝐻_𝑖+𝑠+1𝐴₁𝑃_𝑖𝐵₁+𝑅^𝐻_𝑖+𝑠+1𝐴₂𝑃_𝑖𝐵₂+𝐴₃𝑃^𝐻_𝑖𝐵₃𝑅^𝐻_𝑖+𝑠+1+𝐴₄𝑃^𝑇_𝑖𝐵₄𝑅^𝐻_𝑖+𝑠+1+𝑅_𝑖+𝑠+1²𝑅_𝑖+𝑠²𝑃^𝐻_𝑖+𝑠𝑃_𝑖+𝑅^𝐻_𝑖+𝑠+1𝐶₁𝑄_𝑖𝐷₁+𝑅^𝐻_𝑖+𝑠+1𝐶₂𝑄_𝑖𝐷₂+𝐶₃𝑄^𝐻_𝑖𝐷₃𝑅^𝐻_𝑖+𝑠+1+𝐶₄𝑄^𝑇_𝑘𝐷₄𝑅^𝐻_𝑖+𝑠+1+𝑅_𝑖+𝑠+1²𝑅_𝑖+𝑠²𝑄^𝐻_𝑖+𝑠𝑄_𝑖=Retr𝑅^𝐻_𝑖+𝑠+1𝐴₁𝑃_𝑖𝐵₁+𝐶₁𝑄_𝑖𝐷₁+𝐴₂𝑃_𝑖𝐵₂+𝐶₂𝑄_𝑖𝐷₂+𝐴₃𝑃^𝐻_𝑖𝐵₃+𝐶₃𝑃^𝐻_𝑖𝐷₃+𝐴₄𝑃^𝑇_𝑖𝐵₄+𝐶₄𝑄^𝑇_𝑖𝐷₄+𝑅_𝑖+𝑠+1²𝑅_𝑖+𝑠²𝑃^𝐻_𝑖+𝑠𝑃_𝑖+𝑄^𝐻_𝑖+𝑠𝑄_𝑖=𝑃_𝑖²+𝑄_𝑖²𝑅_𝑖²Retr𝑅^𝐻_𝑖+𝑠+1𝑅_𝑖−𝑅_𝑖+1=𝑃_𝑖²+𝑄_𝑖²𝑅_𝑖²Retr𝑅^𝐻_𝑖+𝑠+1𝑅_𝑖. (28) Also from (19) we have Retr𝑅^𝐻_𝑖+𝑠+1𝑅_𝑖=Retr𝑅_𝑖+𝑠−𝑅_𝑖+𝑠²𝑃_𝑖+𝑠²+𝑄_𝑖+𝑠²×𝐴₁𝑃_𝑖+𝑠𝐵₁+𝐶₁𝑄_𝑖+𝑠𝐷₁+𝐴₂𝑃_𝑖+𝑠𝐵₂+𝐶₂𝑄_𝑖+𝑠𝐷₂+𝐴₃𝑃^𝐻_𝑖+𝑠𝐵₃+𝐶₃𝑄^𝐻_𝑖+𝑠𝐷₃+𝐴₄𝑃^𝑇_𝑖+𝑠𝐵₄+𝐶₄𝑄^𝑇_𝑖+𝑠𝐷₄^𝐻𝑅_𝑖=Retr𝑅^𝐻_𝑖+𝑠𝑅_𝑖−𝑅_𝑖+𝑠²𝑃_𝑖+𝑠²+𝑄_𝑖+𝑠²×tr𝑃^𝐻_𝑖+𝑠𝐴^𝐻₁𝑅_𝑖𝐵^𝐻₁+𝑄^𝐻_𝑖+𝑠𝐶^𝐻₁𝑅_𝑖𝐷^𝐻₁+𝑃^𝐻_𝑖+𝑠𝐴^𝐻₂𝑅_𝑖𝐵^𝐻₂+𝑄^𝐻_𝑖+𝑠𝐶^𝐻₂𝑅_𝑖𝐷^𝐻₂+𝑃_𝑖+𝑠𝐴^𝐻₃𝑅_𝑖𝐵^𝐻₃+𝑄_𝑖+𝑠𝐶^𝐻₃𝑅_𝑖𝐷^𝐻₃+𝑃_𝑖+𝑠𝐴^𝐻₄𝑅_𝑖𝐵^𝐻₄+𝑄_𝑖+𝑠𝐶^𝐻₄𝑅_𝑖𝐷^𝐻₄,Retr𝑅^𝐻_𝑖+𝑠+1𝑅_𝑖𝑅=−^𝑖+𝑠2𝑃_𝑖+𝑠²+𝑄_𝑖+𝑠²×Retr𝑃^𝐻_𝑖+𝑠𝐴^𝐻₁𝑅_𝑖𝐵^𝐻₁+𝑄^𝐻_𝑖+𝑠𝐶^𝐻₁𝑅_𝑖𝐷^𝐻₁+𝑃^𝐻_𝑖+𝑠𝐴^𝐻₂𝑅_𝑖𝐵^𝐻₂+𝑄^𝐻_𝑖+𝑠𝐶^𝐻₂𝑅_𝑖𝐷^𝐻₂+𝐵₃𝑅^𝐻_𝑖𝐴₃𝑃^𝐻_𝑖+𝑠+𝐷₃𝑅^𝐻_𝑖𝐶₃𝑄^𝐻_𝑖+𝑠+𝐵₄𝑅^𝑇_𝑖𝐴₄𝑃^𝐻_𝑖+𝑠+𝐷₄𝑅^𝑇_𝑖𝐶₄𝑄^𝐻_𝑖+𝑠𝑅=−^𝑖+𝑠2𝑃_𝑖+𝑠²+𝑄_𝑖+𝑠²×Retr𝑃^𝐻_𝑖+𝑠𝐴^𝐻₁𝑅_𝑖𝐵^𝐻₁+𝐴^𝐻₂𝑅_𝑖𝐵^𝐻₂+𝐵₃𝑅^𝐻_𝑖𝐴₃+𝐵₄𝑅^𝑇_𝑖𝐴₄+𝑄^𝐻_𝑖+𝑠𝐶^𝐻₁𝑅_𝑖𝐷^𝐻₁+𝐶^𝐻₂𝑅_𝑖𝐷^𝐻₂+𝐷₃𝑅^𝐻_𝑖𝐶₃+𝐷₄𝑅^𝑇_𝑖𝐶₄𝑅=−^𝑖+𝑠2𝑃_𝑖+𝑠²+𝑄_𝑖+𝑠²×Retr𝑃^𝐻_𝑖+𝑠𝑃_𝑖−𝑅_𝑖²𝑅_𝑖−1²𝑃_𝑖−1+𝑄^𝐻_𝑖+𝑠𝑄_𝑖−𝑅_𝑖²𝑅_𝑖−1²𝑄_𝑖−1=𝑅_𝑖+𝑠²𝑃_𝑖+𝑠²+𝑄_𝑖+𝑠²𝑅_𝑖²𝑅_𝑖−1²×Retr𝑃^𝐻_𝑖+𝑠𝑃_𝑖−1+𝑄^𝐻_𝑖+𝑠𝑄_𝑖−1. (29) Repeating (28) and (29), one can easily obtain for certain 𝛼 and 𝛽tr𝑃^𝐻_𝑖+𝑠+1𝑃_𝑖+𝑄^𝐻_𝑖+𝑠+1𝑄_𝑖=𝛼tr𝑃^𝐻_𝑠+1𝑃₁+𝑄^𝐻_𝑠+1𝑄₁,tr𝑅^𝐻_𝑖+𝑠+1𝑅_𝑖=𝛽tr𝑅^𝐻_𝑠+1𝑅₁. (30) Combining these two relations with (26) implies that (24) holds for =𝑠+1. From Steps 1 and 2 the conclusion holds by the principle of induction. With the above two lemmas, we have the following theorem.

Theorem 6 (see [32]). If the matrix equation (1) is consistent, then a solution can be obtained within finite iteration steps by using Algorithm 3 for any initial matrices 𝑉₁, 𝑊₁.

4. Numerical Example

In this section, we present numerical example to illustrate the application of our proposed methods.

Example 7. In this example we illustrate our theoretical results of Algorithm 3 for solving the system of matrix equation: 𝐴₁𝑉𝐵₁+𝐶₁𝑊𝐷₁+𝐴₂𝑉𝐵₂+𝐶₂𝑊𝐷₂+𝐴₃𝑉^𝐻𝐵₃+𝐶₃𝑊^𝐻𝐷₃+𝐴₄𝑉^𝑇𝐵₄+𝐶₄𝑊^𝑇𝐷₄=𝐸. (31) Because of the influence of the error of calculation, the residual 𝑅(𝑘) is usually unequal to zero in this process of the iteration. We regard the matrix 𝑅(𝑘) as a zero matrix if 𝑅(𝑘)<10⁻¹⁰.
Given 𝐴₁=,𝐴2+3𝑖−𝑖1+𝑖51+2𝑖−3₂=,𝐴2+3𝑖−𝑖1+𝑖51+2𝑖−3₃=02−𝑖𝑖−1+3𝑖20,𝐴₄=,𝐶01−3𝑖1+𝑖04+𝑖−3𝑖₁=1+2𝑖3−𝑖4−𝑖2𝑖−3,𝐶₂=,𝐶3+2𝑖01+𝑖04𝑖1−2𝑖₃=1−3𝑖2𝑖−3𝑖12+3𝑖4𝑖,𝐶₄=,𝐵1−2𝑖023−𝑖1+𝑖−1₁=4+𝑖−𝑖01−𝑖4𝑖2+2𝑖,𝐵₂=,𝐵0𝑖1+𝑖0−1−𝑖3𝑖₃=01−3𝑖4+𝑖51+2𝑖,𝐵₄=,𝐷3+𝑖−1−𝑖02−𝑖−1+𝑖2₁=001−3𝑖−𝑖2𝑖−3𝑖,𝐷₂=,𝐷0𝑖1+𝑖0−1−𝑖3𝑖₃=01−3𝑖4+𝑖51+2𝑖,𝐷₄=,.3𝑖−2+𝑖0𝑖−2𝑖−4𝑖𝐸=42+55𝑖115+25𝑖−38−𝑖132+44𝑖 (32) Taking 𝑉₁=⁰⁰⁰⁰₀₀ and 𝑊₁=⁰⁰₀₀ we apply Algorithm 3 to compute 𝑉_𝑘, 𝑊_𝑘.
And iterating 42 steps we get 𝑉=,𝑊=0.0126+1.8415𝑖0.0827+0.6381𝑖1.1221−0.9428𝑖−0.6903+1.0185𝑖1.8818+1.0203𝑖0.9208+0.4569𝑖0.5344−0.4909𝑖0.9280+0.7169𝑖0.4872−0.2734𝑖0.4218−0.9710𝑖0.1763+0.5183𝑖1.1331−0.0432𝑖0.6273−0.1216𝑖−0.3902+0.4313𝑖0.6240+0.9828𝑖−0.7011−0.3418𝑖0.3695+1.7627𝑖−0.3032+0.7073𝑖 (33) which satisfy the matrix equation: 𝐴₁𝑉𝐵₁+𝐶₁𝑊𝐷₁+𝐴₂𝑉𝐵₂+𝐶₂𝑊𝐷₂+𝐴₃𝑉^𝐻𝐵₃+𝐶₃𝑊^𝐻𝐷₃+𝐴₄𝑉^𝑇𝐵₄+𝐶₄𝑊^𝑇𝐷₄=𝐸. (34)
With the corresponding residual 𝑅₄₂=𝐸−𝐴₁𝑉₄₂𝐵₁−𝐶₁𝑊₄₂𝐷₁−𝐴₂𝑉₄₂𝐵₂−𝐶₂𝑊₄₂𝐷₂−𝐴₃𝑉^𝐻₄₂𝐵₃−𝐶₃𝑊^𝐻₄₂𝐷₃−𝐴₄𝑉^𝑇₄₂𝐵₄−𝐶₄𝑊^𝑇₄₂𝐷₄=6.6115×10⁻¹¹. (35)

5. Conclusions

The above Figure 1 shows the convergence curve for the residual function 𝑅(𝑘). In this paper, an iterative algorithm constructed to solve a complex matrix equation with conjugate and transpose of two unknowns of the form: 𝐴₁𝑉𝐵₁+𝐶₁𝑊𝐷₁+𝐴₂𝑉𝐵₂+𝐶₂𝑊𝐷₂+𝐴₃𝑉^𝐻𝐵₃+𝐶₃𝑊^𝐻𝐷₃ + 𝐴₄𝑉^𝑇𝐵₄+𝐶₄𝑊^𝑇𝐷₄=𝐸 is presented. We proved that the iterative algorithms always converge to the solution for any initial matrices. We stated and proved some lemmas and theorems where the solutions are obtained. The proposed method is illustrated by numerical example where the obtained numerical results show that our technique is very neat and efficient.

Figure 1 

The relation between the number of iterations and residual for the example.