The inverse eigenvalue problem of generalized anti-reflexive matrices

A real symmetric unipotent matrix P is said to be generalized 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. This paper involves related inverse eigenvalue problems of generalized anti reflexive matrices and their optimal approximation. Necessary and sufficient conditions for the solvability of the problem are derived, the general expression of the solution is given. The optimal approximate solution is also provided.