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

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 $\tilde{A}(z)$ has a canonical factorization $\tilde{A}(z)=\tilde{U}(z)\tilde{L}(z^{-1})$ and we provide explicit expressions of the factors $\tilde{U}(z)$ and $\tilde{L}(z)$. Similar conditions and expressions are given for the factorization of $\tilde{A}(z^{-1})$. Some applications are discussed.

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