An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations

Abstract

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations $(AXB-CYD,EXF-GYH)=(M,N)$, which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matrices $X$ and $Y$. When the matrix equations are consistent, for any initial generalized reflexive matrix pair $[X_1,Y_1]$, the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors, and the least Frobenius norm generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair $[\widehat{X},\widehat{Y}]$ to a given matrix pair $[X_0,Y_0]$ in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair $[{\tilde{X}}^{*},{\tilde{Y}}^{*}]$ of a new corresponding generalized coupled Sylvester matrix equation pair $(A\tilde{X}B-C\tilde{Y}D,E\tilde{X}F-G\tilde{Y}H)=(\tilde{M},\tilde{N})$, where $\tilde{M}=M-A{X}_{0}B+C{Y}_{0}D,\tilde{N}=N-E{X}_{0}F+G{Y}_{0}H$. Several numerical examples are given to show the effectiveness of the presented iterative algorithm.