Spectral perturbation by rank one matrices

TL;DR

This paper characterizes the conditions under which adding a rank one matrix to a given matrix results in a specified characteristic polynomial, advancing understanding of spectral perturbations.

Contribution

It provides necessary and sufficient conditions for spectral perturbations of matrices by rank one matrices to achieve a desired characteristic polynomial.

Findings

Derived explicit conditions for spectral changes via rank one perturbations.

Extended spectral perturbation theory to arbitrary matrices over algebraically closed fields.

Clarified the relationship between rank one updates and characteristic polynomial modifications.

Abstract

Let $A$ be a matrix of size $n \times n$ over an algebraically closed field $F$ and $q(t)$ a monic polynomial of degree $n$. In this article, we describe the necessary and sufficient conditions of $q(t)$ so that there exists a rank one matrix $B$ such that the characteristic polynomial of $A+B$ is $q(t)$.