The anti-bisymmetric extremal rank solutions of a linear matrix equation

In this paper, the rank range of the solutions of a class of matrix equation are considered. By applying the singular value decomposition of matrix and the properties of Frobenius matrix norm, the extremal rank and the solution expression are also given when the solvability conditions are satisfied. Then, the explicit expression for the nearest matrix to a given matrix, that belongs to the corresponding minimal rank solution set of the equation, in the Frobenius norm has been provided.