Optimization problems on the rank of the solution to left and right inverse eigenvalue problem

Abstract

A complex matrix $P$ is called Hermitian and $\{k+1\}$-potent if $P^{k+1}=P=P^*$ for some integer $k\geq 1$. Let $P$ and $Q$ be $n\times n$ Hermitian and $\{k+1\}$-potent matrices, we say that complex matrix $A$ is $\{P,Q,k+1\}$-reflexive (anti-reflexive) if $PAQ=A$ ($PAQ=-A$). In this paper, the solvability conditions and the general solutions of the left and right inverse eigenvalue problem for $\{P,Q,k+1\}$-reflexive and anti-reflexive matrices are derived, and the minimal and maximal rank solutions are given. Moreover, the associated optimal approximation problem is also considered. Finally, numerical example is given to illustrate the main results.