Conjugate Matrix Equations Research Articles

A modified gradient‐based iterative algorithm for solving the complex conjugate and transpose matrix equations

In this paper, we first develop the modified gradient‐based iterative (MGI) method for the complex conjugate and transpose matrix equations . By adopting the updated technique, we can make full use of the latest information to compute the next result, which leads to a faster convergence rate. In theory, we apply the real representation of a complex matrix and the vec‐operator to prove the convergence properties. Furthermore, we extend the MGI algorithm to solve the generalized complex conjugate and transpose matrix equations. Then, the necessary and sufficient conditions for convergence of the MGI algorithm are presented. Lastly, three numerical examples are introduced to testify the efficiency of our methods.

Modified and accelerated relaxed gradient-based iterative algorithms for the complex conjugate and transpose matrix equations

Modified and accelerated relaxed gradient-based iterative algorithms for the complex conjugate and transpose matrix equations

The Weighted, Relaxed Gradient-Based Iterative Algorithm for the Generalized Coupled Conjugate and Transpose Sylvester Matrix Equations

By applying the weighted relaxation technique to the gradient-based iterative (GI) algorithm and taking proper weighted combinations of the solutions, this paper proposes the weighted, relaxed gradient-based iterative (WRGI) algorithm to solve the generalized coupled conjugate and transpose Sylvester matrix equations. With the real representation of a complex matrix as a tool, the necessary and sufficient conditions for the convergence of the WRGI algorithm are determined. Also, some sufficient convergence conditions of the WRGI algorithm are presented. Moreover, the optimal step size and the corresponding optimal convergence factor of the WRGI algorithm are given. Lastly, some numerical examples are provided to demonstrate the effectiveness, feasibility and superiority of the proposed algorithm.

A NOVEL ITERATIVE METHOD FOR SOLVING THE COUPLED SYLVESTER-CONJUGATE MATRIX EQUATIONS AND ITS APPLICATION IN ANTILINEAR SYSTEM

This paper is devoted to constructing a modified relaxed gradient based iterative (MRGI) algorithm to solve the coupled Sylvester-conjugate matrix equations (CSCMEs) based on the hierarchical identification principle. Convergence analysis shows that the proposed algorithm is effective for arbitrary initial matrices. Further, we apply the MRGI algorithm to study a more general coupled Sylvester conjugate matrix equations and give a sufficient condition to guarantee that the iterative solution converges to the exact solution. Two numerical experiments are provided to demonstrate that the MRGI algorithm has better efficiency and accuracy than the three existing algorithms, which are presented by Wu et al. (2010) and Huang and Ma (2018). Finally, we derive an application of MRGI algorithm in discrete-time antilinear system.

Iterative solution to a class of complex matrix equations and its application in time-varying linear system

Wu et al. (Applied Mathematics and Computation 217(2011)8343-8353) constructed a gradient based iterative (GI) algorithm to find the solution to the complex conjugate and transpose matrix equation $$\begin{aligned} A_{1}XB_{1}+A_{2}\overline{X}B_{2}+A_{3}X^{T}B_{3}+A_{4}X^{H}B_{4}=E \end{aligned}$$ and a sufficient condition for guaranteeing the convergence of GI algorithm was given for an arbitrary initial matrix. Zhang et al. (Journal of the Franklin Institute 354 (2017) 7585-7603) provided a new proof of GI method and the necessary and sufficient conditions was presented to guarantee that the proposed algorithm was convergent for an arbitrary initial matrix. In this paper, a relaxed gradient based iterative (RGI) algorithm is proposed to solve this complex conjugate and transpose matrix equation. The necessary and sufficient conditions for the convergence factor is determined to guarantee the convergence of the introduced algorithm for any initial iterative matrix. Numerical results are given to verify the efficiency of the new method. Finally, the application in time-varying linear system of the presented algorithm is provided.

Gradient-Based Iterative Algorithm for a Coupled Complex Conjugate and Transpose Matrix Equations

Gradient-based iterative algorithm is suggested for solving a coupled complex conjugate and transpose matrix equations. Using the hierarchical identification principle and the real representation of a complex matrix, a convergence proof is offered. The necessary and sufficient conditions for the optimal convergence factor are determined. A numerical example is offered to validate the efficacy of the suggested algorithm.

Consistency of Split Quaternion Matrix Equations $$AX^{\star }-XB=CY+D$$ and $$X-AX^\star B=CY+D$$

In this paper, the split quaternion matrix equations $$AX^{\star }-XB=CY+D$$ and $$X-AX^{\star }B=CY+D$$ are considered, where $$X^\star $$ is X or the $$\phi $$ -conjugate of X, $$\phi \in \{i, j, k\}$$ . We design some new real representations of a split quaternion matrix, which enable us to convert $$\phi $$ -conjugate matrix equations into some real matrix equations. By using this method, we show that the original split quaternion matrix equation is consistent if and only if its corresponding real matrix equation is consistent. Moreover, we formulate the split quaternion solutions by the real solutions of its corresponding real matrix equation. As applications, we derive the necessary and sufficient conditions for the existence of solutions to the above two split quaternion matrix equations and their special cases $$AX^{\star }-XB=C$$ and $$X-AX^{\star }B=C$$ . We also obtain some conditions for the mentioned four split matrix equations to be uniquely solvable.

Solutions of a second-order conjugate matrix equation

ABSTRACT The following second-order conjugate matrix equation is considered where are known matrices and is an unknown matrix to be determined. The necessary and sufficient condition for the matrix equation to have a solution is established and the explicit solutions of the equation are provided.

Consistency of Quaternion Matrix Equations $AX^{\star}-XB=C$ and $X-AX^\star B=C$

For a given ordered units triple $\{q_1, q_2, q_3\}$, the solutions to the quaternion matrix equations $AX^{\star}-XB=C$ and $X-AX^{\star}B=C$, $X^{\star} \in \{ X , X^{\eta} , X^* , X^{\eta*}\}$, where $X^*$ is the conjugate transpose of $X$, $X^{\eta}=-\eta X \eta$ and $X^{\eta*}=-\eta X^* \eta$, $\eta \in \{q_1, q_2, q_3\}$, are discussed. Some new real representations of quaternion matrices are used, which enable one to convert $\eta$-conjugate (transpose) matrix equations into some real matrix equations. By using this idea, conditions for the existence and uniqueness of solutions to the above quaternion matrix equations are derived. Also, methods to construct the solutions from some related real matrix equations are presented.

Finite iterative Hamiltonian solutions of the generalized coupled Sylvester – conjugate matrix equations

In this paper, we present an iterative algorithm to solve a generalized coupled Sylvester – conjugate matrix equations over Hamiltonian matrices. When the considered systems of matrix equations are consistent, it is proven that the solution can be obtained within finite iterative steps for any arbitrary initial generalized Hamiltonian matrices in the absence of round off errors. Two numerical examples are given to illustrate the effectiveness of the proposed method.

New proof of the gradient-based iterative algorithm for a complex conjugate and transpose matrix equation

By using the hierarchical identification principle, the gradient-based iterative algorithm is suggested for solving a complex conjugate and transpose matrix equation. With the tools of the real representation of a complex matrix and the vec-operator, a new convergence proof is offered. The necessary and sufficient conditions for the convergence factor is determined to guarantee the convergence of the algorithm for any initial iterative matrix. Also a conjecture proposed by Wu et al. (2011) is solved. Two numerical examples are offered to illustrate the effectiveness of the suggested algorithm and conclusions.

New proof of the gradient-based iterative algorithm for the Sylvester conjugate matrix equation

By applying the hierarchical identification principle, the gradient-based iterative algorithm is suggested for solving the Sylvester conjugate matrix equation. With the real representation of a complex matrix, a new convergence proof is given. The necessary and sufficient conditions for the convergence factor is determined to guarantee the convergence of the algorithm for any initial iterative matrix. Also a conjecture by Wu et al. (2010) is solved. A numerical example is offered to illustrate the effectiveness of the suggested algorithm and verify some conclusions proposed in this paper.

An accelerated gradient-based iterative algorithm for solving extended Sylvester–conjugate matrix equations

In this paper, we present an accelerated gradient-based iterative algorithm for solving extended Sylvester–conjugate matrix equations. The idea is from the gradient-based method introduced in Wu et al. ( Applied Mathematics and Computation 217(1): 130–142, 2010a) and the relaxed gradient-based algorithm proposed in Ramadan et al. ( Asian Journal of Control 16(5): 1–8, 2014) and the modified gradient-based algorithm proposed in Bayoumi (PhD thesis, Ain Shams University, 2014). The convergence analysis of the algorithm is investigated. We show that the iterative solution converges to the exact solution for any initial value provided some appropriate assumptions be made. A numerical example is given to illustrate the effectiveness of the proposed method and to test its efficiency and accuracy compared with an existing one presented in Wu et al. (2010a), Ramadan et al. (2014) and Bayoumi (2014).

Real iterative algorithms for a common solution to the complex conjugate matrix equation system

In this paper, we propose a new algorithm for the computation of the common solution to the complex conjugate matrix equation system AiXBi+CiX¯Di=Ei,i=1,2,…,N. The algorithms only need finite iteration steps, requiring only real computation. Furthermore, the algorithm can be extended to solve a more general complex matrix equation system. Two numerical examples are given to illustrate the effectiveness of the proposed algorithm.

Gradient-based iterative algorithm for solving the generalized coupled Sylvester-transpose and conjugate matrix equations over reflexive (anti-reflexive) matrices

Linear matrix equations play an important role in many areas, such as control theory, system theory, stability theory and some other fields of pure and applied mathematics. In the present paper, we consider the generalized coupled Sylvester-transpose and conjugate matrix equations [Formula: see text] where [Formula: see text] is a group of unknown matrices and for [Formula: see text], [Formula: see text] in which [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are given matrices with suitable dimensions defined over complex number field. By using the hierarchical identification principle, an iterative algorithm is proposed for solving the above coupled linear matrix equations over the group of reflexive (anti-reflexive) matrices. Meanwhile, sufficient conditions are established which guarantee the convergence of the presented algorithm. Finally, some numerical examples are given to demonstrate the validity of our theoretical results and the efficiency of the algorithm for solving the mentioned coupled linear matrix equations.

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

We consider an iterative algorithm for solving a complex matrix equation with conjugate and transpose of two unknowns of the form: A1VB1+C1WD1+A2V¯B2+C2W¯D2+A3VHB3+C3WHD3+A4VTB4 + C4WTD4=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.

Iterative algorithms for solving a class of complex conjugate and transpose matrix equations

This paper is concerned with iterative solutions to a class of complex matrix equations, which include some previously investigated matrix equations as special cases. By applying the hierarchical identification principle, an iterative algorithm is constructed to solve this class of matrix equations. A sufficient condition is presented to guarantee that the proposed algorithm is convergent for an arbitrary initial matrix with a real representation of a complex matrix as tools. By using some properties of the real representation, a convergence condition that is easier to compute is also given in terms of original coefficient matrices. A numerical example is employed to illustrate the effectiveness of the proposed methods.