The Structure of Symmetric Solutions of the Matrix Equation AX = B over a Principal Ideal Domain

We investigate the structure of symmetric solutions of the matrix equation A X = B , where A and B are m ‐by‐ n matrices over a principal ideal domain R and X is unknown n ‐by‐ n matrix over R. We prove that matrix equation A X = B over R has a symmetric solution if and only if equation A X = B has a solution over R and the matrix A B T is symmetric. If symmetric solution exists we propose the method for its construction.