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

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.