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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'stheorem in functional form and provide extensions to matrix polynomials and tomatrix Laurent series $A(z)$ together with generalizations to shifting a set ofeigenvalues. We provide conditions under which the modified function$\widetilde A(z)$ has a canonical factorization $\widetilde A(z)=\widetildeU(z)\widetilde L(z^{-1})$ and we provide explicit expressions of the factors$\widetilde U(z)$ and $\widetilde L(z)$. Similar conditions and expressions aregiven for the factorization of $\widetilde A(z^{-1})$. Some applications arediscussed.

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