On the relation between the numerical range and the joint numerical range of matrix polynomials

Abstract

It is shown that the numerical range, NR[P ( )], of a matrix polynomial P ( ) = Am m + : : :+A1 +A0 consists of the roots of all scalar polynomials whose coeAEcients correspond to the elements of the convex hull of the joint numerical range of the (m+1)-tuple (A0; A1; : : : ; Am). Moreover, the elements of the joint numerical range that give rise to scalar polynomials with a common root belonging to NR[P ( )] form a connected set. The latter fact is used to examine the multiplicity of roots belonging to the intersection of the root zones of NR[P ( )]. Also an approximation scheme for NR[P ( )] is proposed, in terms of numerical ranges of diagonal matrix polynomials.