SOLUTION OF EQUATION AX + XB = C BY INVERSION OF AN M × M OR N × N MATRIX ∗

It is often of interest to solve the equation AX + XB = C (1) for X, where X and C are M × N real matrices, A is an M × M real matrix, and B is an N × N real matrix. A familiar example occurs in the Lyapunov theory of stability [1], [2], [3] with B = A . Is also arises in the theory of structures [4]. Using the notation P ×Q to denote the Kronecker product (PijQ) (see [5]) in which each element of P is multipled by Q, we find that the equation written out in full for the MN unknowns x11, x21, . . . , x12, . . . in terms of c11, c21, . . . , c12, . . . becomes