On the Stability Radius of Matrix Polynomials

Abstract

The stability radius of a matrix polynomial P ( λ ) relative to an open region ω of the complex plane and its relation to the numerical range of P ( λ ) are investigated. Using an expression of the stability radius in terms of λ on the boundary of ω and Á P ( λ ) m 1 Á 2 , a lower bound is obtained. This bound for the stability radius involves the distances of ω to the connected components of the numerical range of P ( λ ) and can be applied in conjunction with polygonal approximations of the numerical range. The special case of hyperbolic matrix polynomials is also considered.