Generalization and variations of Pellet's theorem for matrix polynomials

Aaron Melman

Published 2012-09-30, updated 2013-02-15Version 2

We derive a generalized matrix version of Pellet's theorem, itself based on a generalized Rouch\'{e} 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.