An inverse eigenproblem for generalized reflexive matrices with normal $k+1$-potencies

Let P, Q ∈ C be two normal {k+1}-potent matrices, i.e., PP ∗ = P P, P k+1 = P , QQ = QQ, Q = Q, k ∈ N. A matrix A ∈ C is referred to as generalized reflexive with two normal {k + 1}-potent matrices P and Q if and only if A = PAQ. The set of all n × n generalized reflexive matrices which rely on the matrices P and Q is denoted by GR(P,Q). The left and right inverse eigenproblem of such matrices ask from us to find a matrix A ∈ GR(P,Q) containing a given part of left and right eigenvalues and corresponding left and right eigenvectors. In this paper, first necessary and sufficient conditions such that the problem is solvable are obtained. A general representation of the solution is presented. Then an expression of the solution for the optimal Frobenius norm approximation problem is exploited. A stability analysis of the optimal approximate solution, which has scarcely been considered in existing literature, is also developed.