The general solutions on the minimum residual problem and the matrix nearness problem for symmetric matrices or anti-symmetric matrices

Abstract

This paper gives the precise solutions on the minimum residual problem and the matrix nearness problem for symmetric matrices or anti-symmetric matrices. Firstly,an equivalent linear equation is given for the minimum residual problem for symmetric matrices. Then, by generalized singular value decomposition (GSVD) the expression of a general solution on the minimum residual problem for symmetric matrices is arrived. Furthermore, for the matrix nearness problem for symmetric matrices, an unique optimal approximate solution is given by first find the general symmetric solution of a corresponding equivalent matrix equation of the matrix nearness problem. In addition, explicit solutions on the minimum residual problem and the matrix nearness problem for anti-symmetric matrices are derived.