An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation Mehdi Dehghan * , Masoud Hajarian Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Avenue, Tehran 15914, Iran Abstract The generalized coupled Sylvester matrix equations ðAY ZB; CY ZDÞ¼ðE; F Þ with unknown matrices Y ; Z are encountered in many systems and control applications. Also these matrix equations have several applications relating to the problem of computing stable eigendecompositions of matrix pencils. In this work, we construct an iterative algo- rithm to solve the generalized coupled Sylvester matrix equations over reflexive matrices Y ; Z. And when the matrix equa- tions are consistent, for any initial matrix pair ½Y 0 ; Z 0 , a reflexive solution pair can be obtained within finite iteration steps in the absence of roundoff errors, and the least Frobenius norm reflexive solution pair can be obtained by choosing a spe- cial kind of initial matrix pair. Also we obtain the optimal approximation reflexive solution pair to a given matrix pair ½ Y ; Z in the reflexive solution pair set of the generalized coupled Sylvester matrix equations ðAY ZB; CY ZDÞ¼ðE; F Þ. Moreover, several numerical examples are given to show the efficiency of the presented iterative algorithm. Ó 2008 Elsevier Inc. All rights reserved. Keywords: The generalized coupled Sylvester matrix equations; Generalized reflection matrix; Kronecker matrix product; Reflexive matrix; Optimal approximation reflexive solution pair 1. Introduction We first give some notations which are used in this paper. The notation R mn denotes the set of all m n real matrices. The unit matrix of order n is denoted by I n .1 n denotes the matrix of order n whose all elements are 1. We use A T , trðAÞ and RðAÞ to denote the transpose, the trace and the column space of the matrix A, respectively. For a matrix A 2 R mn , the so–called stretching function vecðAÞ is defined by the following: vecðAÞ¼ a T 1 a T 2 ... a T n T ; 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.02.035 * Corresponding author. E-mail addresses: mdehghan@aut.ac.ir (M. Dehghan), mhajarian@aut.ac.ir, masoudhajarian@gmail.com (M. Hajarian). Available online at www.sciencedirect.com Applied Mathematics and Computation 202 (2008) 571–588 www.elsevier.com/locate/amc