Generalization and variations of Pellet's theorem for matrix polynomials
arXiv:1210.017238 citationsh-index: 13
Originality Synthesis-oriented
AI Analysis
Provides new theoretical tools for bounding eigenvalues of matrix polynomials, benefiting numerical linear algebra and control theory, though incremental in nature.
The authors generalize Pellet's theorem to matrix polynomials, enabling eigenvalue bounds where the classical theorem fails. They provide upper, lower, and internal bounds, with variations to handle previously inapplicable cases.
We derive a generalized matrix version of Pellet's theorem, itself based on a generalized Rouché theorem for matrix-valued functions, to generate upper, lower, and internal bounds on the eigenvalues of matrix polynomials. Variations of the theorem are suggested to try and overcome situations where Pellet's theorem cannot be applied.