%0 Journal Article
%T Finite Iterative Algorithm for Solving a Complex of Conjugate and Transpose Matrix Equation
%A Mohamed A. Ramadan
%A Talaat S. El-Danaf
%A Ahmed M. E. Bayoumi
%J Journal of Discrete Mathematics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/170263
%X We consider an iterative algorithm for solving a complex matrix equation with conjugate and transpose of two unknowns of the form: + . 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 , 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: where , , , and are given matrices, while 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 can be characterized by the consimilarity [11¨C13] of two partitioned matrices related to the coefficient matrices , and . By consimilarity Jordan decomposition, explicit solutions were obtained in [10, 14]. Some explicit expressions of the solution to the matrix
%U http://www.hindawi.com/journals/jdm/2013/170263/