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: A(i) is an element of C-piXti B-i is an element of C-siXqi Ci is an element of CpiXti+1, D-i is an element of Csi+1Xqi, and E-i is an element of C-piXqi (i = 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 A(i)X(i)B(i) + CiXi+1Di = Ei (i = 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 A(i), B-i, C-i, D-i, and E-i. 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 + 1 unknowns A(i)X(i)B(i) + CiXi+1Di = E-i (i = 1, . . ., k, k >= 4).