On the matrix equations U_iXV_j W_{i j} for 1 \leq i; j \leq k with i +j \leq k

Conditions for the existence of a common solution X for the linear matrix equations U_iXV_j 􏰁 W_{ij} for 1 \leq 􏰃 i,j \leq 􏰂 k with i\leq 􏰀 j \leq 􏰃 k, where the given matrices U_i,V_j,W_{ij} and the unknown matrix X have suitable dimensions, are derived. Verifiable necessary and sufficient solvability conditions, stated directly in terms of the given matrices and not using Kronecker products, are also presented. As an application, a version of the almost triangular decoupling problem is studied, and conditions for its solvability in transfer matrix and state space terms are presented.