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

Abstract

Let $P,~Q\in\mathbb{C}^{n\times n}$ be two normal $\{k+1\}$-potent matrices, i.e., $PP^{*}=P^{*}P,~P^{k+1}=P$, $QQ^{*}=Q^{*}Q,~Q^{k+1}=Q$, $k\in\mathbb{N}$. A matrix $A\in\mathbb{C}^{n\times n}$ 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 $n\times n$ generalized reflexive matrices which rely on the matrices $P$ and $Q$ is denoted by $\mathcal{GR}^{n\times n}(P,Q)$. The left and right inverse eigenproblem of such matrices ask from us to find a matrix $A\in\mathcal{GR}^{n\times n}(P,Q)$ 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.