The solution of the equation AX+X*B=0

We describe how to find the general solution of the matrix equation AX + (XB)-B-star = 0, where A is an element of C-mxn and B is an element of C-nxm are arbitrary matrices, X is an element of C-nxm is the unknown, and X-star 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 + lambda B-star 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 + lambda B-star 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 + lambda B-star. The general solution of the homogeneous equation AX + (XB)-B-star = 0 is essential to finding the general solution of AX + (XB)-B-star = C, which is related to palindromic eigenvalue problems that have attracted considerable attention recently.

The solution of the equation AX+X*B=0 Articles