Volume 17, pp. 1-10, 2004.

On the estimation of the $q$-numerical range of monic matrix polynomials

Panayiotis J. Psarrakos

Abstract

For a given $q\in [0,1],$ the $q$-numerical range of an $n\times n$ matrix polynomial $P(\lambda )=I{\lambda }^m+A_{m-1}{\lambda }^{m-1}+\cdots +A_1\lambda +A_0$ is defined by $W_q(P)=\{\lambda \in \mathbf{C}:y^{\ast }P(\lambda )x=0,x,y\in {\mathbf{C}}^n,x^{\ast }x=y^{\ast }y=1,y^{\ast }x=q\}$. In this paper, an inclusion-exclusion methodology for the estimation of $W_q(P)$ is proposed. Our approach is based on i) the discretization of a region $\mathrm{\Omega }$ that contains $W_q(P)$, and ii) the construction of an open circular disk, which does not intersect $W_q(P)$, centered at every grid point $\mu \in \mathrm{\Omega }\setminus W_q(P)$. For the cases $q=1$ and $0<q<1,$ an important difference arises in one of the steps of the algorithm. Thus, these two cases are discussed separately.

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Key words

matrix polynomial, eigenvalue, $q$-numerical range, boundary, inner $q$-numerical radius, Davis-Wielandt shell.

AMS subject classifications

15A22,15A60,65D18,65F30,65F35.