Consistency and Efficient Solution of the Sylvester

We consider the matrix equation AX + X⋆B = C , where the matrices A and B have sizes m × n and n × m, respectively, the size of the unknown X is n × m, and the operator (·)⋆ denotes either the transpose or the conjugate transpose of a matrix. In the first part of the paper, we review necessary and sufficient conditions for the existence and uniqueness of solutions. These conditions were obtained previously by Wimmer and by Byers, Kressner, Schröder and Watkins. In this review, we generalize to fields of characteristic different from two the existence condition that Wimmer originally proved for the complex field. In the second part, we develop, in the real or complex square case m = n, an algorithm to solve the equation in O(n3) flops when the solution is unique. This algorithm is based on the generalized Schur decomposition of the matrix pencil A− λB⋆. The equation AX + X⋆B = C is connected with palindromic eigenvalue problems and, as a consequence, the square complex case has attracted recently the attention of several authors. For instance, Byers, Kressner, Schröder and Watkins have considered this equation in the study of the conditioning and in the development of structured algorithms for palindromic eigenvalue problems, while De Terán and Dopico have solved the homogeneous equations AX +X⋆A = 0 and studied their relationship with orbits of matrices and palindromic pencils under the action of ⋆-congruence.