THE ANTI-CENTRO-SYMMETRIC EXTREMAL RANK SOLUTIONS OF THE MATRIX EQUATION AX = B

A matrix A = (a_ij) 2 ϵ R^nn is said to be a centro-symmetric matrix if a_ij = -a_n+i1-n+1 ij= 1; 2... n. In this paper, we mainly investigate the anti-centro-symmetric maximal and minimal rank solutions to the system of matrix equation AX = B. We present necessary and sufficient conditions for the existence of the maximal and minimal rank solutions with anti-centro-symmetric to the system. The expressions of such solutions to this system are also given when the solvability conditions are satisfied. In addition, in corresponding the minimal rank solution set to the system, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm has been provided.