Computing a Nearest P-Symmetric Nonnegative Definite Matrix Under Linear Restriction

Abstract

Let $P$ be an $n\times n$ symmetric orthogonal matrix. A real $n\times n$ matrix $A$ is called P-symmetric nonnegative definite if $A$ is symmetric nonnegative definite and $(PA)^T=PA$. This paper is concerned with a kind of inverse problem for P-symmetric nonnegative definite matrices: Given a real $n\times n$ matrix $\widetilde{A}$, real $n\times m$ matrices $X$ and $B$, find an $n\times n$ P-symmetric nonnegative definite matrix $A$ minimizing $||A-\widetilde{A}||_F$ subject to $AX =B$. Necessary and sufficient conditions are presented for the solvability of the problem. The expression of the solution to the problem is given. These results are applied to solve an inverse eigenvalue problem for P-symmetric nonnegative definite matrices.