Generalization of the Brauer Theorem to Matrix Polynomials and Matrix Laurent Series
Provides theoretical extensions for eigenvalue modification in matrix functions, relevant to numerical linear algebra and systems theory, but is incremental.
The paper extends Brauer's theorem, which modifies a single eigenvalue of a matrix via rank-one perturbation, to matrix polynomials and Laurent series, enabling shifting sets of eigenvalues while preserving canonical factorizations. Explicit factorizations and conditions are provided.
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.