On the matrix equations U_IXV_J = W_{I J} for 1 ≤ I, J < K with I + J ≤ K

Conditions for the existence of a common solution X for the linear matrix equations U_iXV_j = W_{i j} for 1 ≤ i, j < k with i + j ≤ k, where the given matrices U_i, V_j,W_{i j} and the unknown matrix X have suitable dimensions, are derived. Verifiable necessary and sufficient solvability con-ditions, 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.