On the rank-k numerical range of matrices polynomials

Abstract

This article introduces the notion of the rank-$k$ numerical range $\Lambda_{k}(L)$ of a matrix polynomial $L(\lambda) = A_{m} \lambda^{m} + \cdots + A_{1} \lambda + A_{0}$, whose coefficients are $n\times n$ complex matrices. Also, geometric properties are obtained, including the relation to the ordinary numerical range $W(L)$.