ELA, Volume 6, pp. 20-30, March 2000, abstract.

On the Relation Between the Numerical Range and 
the Joint Numerical Range of Matrix Polynomials

P. J. Psarrakos and M. J. Tsatsomeros

It is shown that the numerical range, NR[P(lambda)], of a matrix 
polynomial P(lambda) = A_m lambda^m + ... + A_1 lambda + A_0
consists of the roots of all scalar polynomials whose coefficients 
correspond to the elements of the convex hull of the joint numerical 
range of the (m+1)-tuple (A_0,A_1,...,A_m).
Moreover, the elements of the joint numerical range that give rise
to scalar polynomials with a common root belonging to NR[P(lambda)]
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(lambda)].  Also an approximation scheme for NR[P(lambda)] is 
proposed, in terms of numerical ranges of diagonal matrix polynomials.