The symmetric ortho-symmetric extremal rank solutions to a class of matrix equation

The symmetric ortho-symmetric maximal and minimal rank solutions to a class of matrix equation and their optimal approximation are considered. By applying the matrix rank method, the necessary and sufficient conditions for the existence of the maximal and minimal rank solutions with symmetric ortho-symmetric to the equation. The expressions of such solutions to this equation are also given when the solvability conditions are satisfied. In addition, in corresponding the minimal rank solution set to the equation, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm has been provided.