The solution of the equation AX+X★B=0

Abstract We describe how to find the general solution of the matrix equation AX + X ★ B = 0 , where A ∈ C m × n and B ∈ C n × m are arbitrary matrices, X ∈ C n × 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 + λ B ★ 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 + λ B ★ 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 + λ B ★ . 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.

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