Left and Right Inverse Eigenpairs Problem of Generalized Anti-reflexive Matrices and Its Approximation

A real symmetric unipotent matrix P is said to be generalized anti-reflection matrix. A real matrix A is said to be a generalized anti-reflexive matrix with respect to generalized reflection matrix dual (P, Q) if A =-PAQ. In this paper, the left and right inverse eigenpairs problems for generalized anti-reflexive matrices are considered. We obtain the necessary and sufficient conditions for the solvability of the problem and we present the general expression of the solution. The related optimal approximation problem to a given matrix over the solution set is solved. In addition, a numerical algorithm and examples to solve the problem are given.