The solution of the equation AX+X★B=0

We describe how to find the general solution of the matrix equation AX+X★B=0, where A∈Cm×n and B∈Cn×m are arbitrary matrices, X∈Cn×m is the unknown, and X★ denotes either the transpose or the conjugate transpose of X. We first show that the solution can be obtained in terms of the Kronecker canonical form of the matrix pencil A+λB★ and the two nonsingular matrices which transform this pencil into its Kronecker canonical form. We also give a complete description of the solution provided that these two matrices and the Kronecker canonical form of A+λB★ are known. As a consequence, we determine the dimension of the solution space of the equation in terms of the sizes of the blocks appearing in the Kronecker canonical form of A+λB★. The general solution of the homogeneous equation AX+X★B=0 is essential to finding the general solution of AX+X★B=C, which is related to palindromic eigenvalue problems that have attracted considerable attention recently.