Extensions of the Eneström-Kakeya theorem for matrix polynomials

Abstract

The classical Eneström-Kakeya theorem establishes explicit upper and lower bounds on the zeros of a polynomial with positive coefficients and has been generalized for positive definite matrix polynomials by several authors. Recently, extensions that improve the (scalar) Eneström-Kakeya theorem were obtained with a transparent and unified approach using just a few tools. Here, the same tools are used to generalize these extensions to positive definite matrix polynomials, while at the same time generalizing the tools themselves. In the process, a framework is developed that can naturally generate additional similar results.

Keywords

• matrix polynomial
• positive definite
• polynomial eigenvalue
• cauchy
• eneström-kakeya
• 12d10
• 30c15
• 65h05