Two-sided coupled generalized Sylvester matrix equations solving using a simultaneous decomposition for fifteen matrices

In this paper, we investigate and analyze in detail the structure and properties of a simultaneous decomposition for fifteen matrices: Ai∈Cpi×tiAi∈Cpi×ti, Bi∈Csi×qiBi∈Csi×qi, Ci∈Cpi×ti+1Ci∈Cpi×ti+1, Di∈Csi+1×qiDi∈Csi+1×qi, and Ei∈Cpi×qiEi∈Cpi×qi (i=1,2,3i=1,2,3). We show that from this simultaneous decomposition we can derive some necessary and sufficient conditions for the existence of a solution to the system of two-sided coupled generalized Sylvester matrix equations with four unknowns AiXiBi+CiXi+1Di=EiAiXiBi+CiXi+1Di=Ei (i=1,2,3i=1,2,3). Apart from proving an expression for the general solutions to this system, we derive the range of ranks of these solutions using the ranks of the given matrices AiAi, BiBi, CiCi, DiDi, and EiEi. We provide some numerical examples to illustrate our results. Moreover, we present a similar approach to consider the simultaneous decomposition for 5k matrices and the system of k   two-sided coupled generalized Sylvester matrix equations with k+1k+1 unknowns AiXiBi+CiXi+1Di=EiAiXiBi+CiXi+1Di=Ei (i=1,…,ki=1,…,k, k≥4k≥4). The main results are also valid over the real number field and the real quaternion algebra.