On the relation between the numerical range and the joint numerical range of matrix polynomials
Panayiotis Psarrakos, Michael J. Tsatsomeros · Electronic Journal of Linear Algebra · 1999
It is shown that the numerical range, NR[P()], of a matrix polynomial P () = Am m + : : : + A 1 + A 0 consists of the roots of all scalar polynomials whose coeÆcients correspond to the elements of the convex hull of the joint n umerical 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()] 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.