Nearest matrix with prescribed eigenvalues and its applications

Abstract

Consider $n\times n$ matrix $A$ and a set $\Lambda$ consisting of $k\le n$ prescribed complex numbers. Lippert (2010) in a challenging article, studied geometrically the spectral norm distance from $A$ to the set $\Lambda$ and constructed a perturbation matrix $\Delta$ with minimum spectral norm such that $A+\Delta$ had $\Lambda$ in its spectrum. This paper presents an easy practical computational method for constructing the optimal perturbation $\Delta$ by extending necessary definitions and lemmas of previous works. Also, some conceivable applications of this issue are provided.