a b s t r a c tIn this paper, we propose a new algorithm to solve the Sylvester matrix equation XA + BX = C . The technique consists of orthogonal reduction of the matrix A to a block upper Hessenberg form P T AP = H and then solving the reduced equation, YH + BY = C for Y through recurrence relation, where Y = XP, and C = CP. We then recover the solution of the original problem via the relation X = YP T . The numerical results show the accuracy and the efficiency of the proposed algorithm. In addition, how the technique described can be applied to other matrix equations was shown.

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In this paper, the generalized Sylvester matrix equation AV + BW = EVF + C over reflexive matrices is considered. An iterative algorithm for obtaining reflexive solutions of this matrix equation is introduced. When this matrix equation is consistent over reflexive solutions then for any initial reflexive matrix, the solution can be obtained within finite iteration steps. Furthermore, the complexity and the convergence analysis for the proposed algorithm are given. The least Frobenius norm reflexive solutions can also be obtained when special initial reflexive matrices are chosen. Finally, numerical examples are given to illustrate the effectiveness of the proposed algorithm.