Generalization of the Brauer Theorem to Matrix Polynomials and Matrix Laurent Series

TL;DR

This paper extends Brauer's eigenvalue modification theorem from matrices to matrix polynomials and Laurent series, providing conditions for factorization and explicit formulas for the modified functions.

Contribution

It reformulates Brauer's theorem in a functional setting and generalizes it to matrix polynomials and Laurent series, including eigenvalue set modifications and factorization conditions.

Findings

Provides explicit factorization formulas for modified matrix functions.

Establishes conditions for canonical factorization after eigenvalue modifications.

Discusses applications of the generalized theorem in matrix analysis.

Abstract

Given a square matrix $A$, Brauer's theorem [Duke Math. J. 19 (1952), 75--91] shows how to modify one single eigenvalue of $A$ via a rank-one perturbation, without changing any of the remaining eigenvalues. We reformulate Brauer's theorem in functional form and provide extensions to matrix polynomials and to matrix Laurent series $A(z)$ together with generalizations to shifting a set of eigenvalues. We provide conditions under which the modified function $\widetilde A(z)$ has a canonical factorization $\widetilde A(z)=\widetilde U(z)\widetilde L(z^{-1})$ and we provide explicit expressions of the factors $\widetilde U(z)$ and $\widetilde L(z)$. Similar conditions and expressions are given for the factorization of $\widetilde A(z^{-1})$. Some applications are discussed.