On minimal solutions of the matrix equation AX−YB=0 Original Research Article

In the case when one of the ranks of matrices A or B is full, all self-adjoint pairs of solutions of the equation AX−YB=0 are given. Necessary and sufficient conditions for the existence of nonnegative and positive definite solutions are proved. Without any condition on ranks and with a given solution X, it is shown that maximal and minimal solutions for Y do not exist in nontrivial cases. It is also proved that the minimal nonnegative solution Y exists. An explicit formula for this solution is given.