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 {\bf C} : y^*P(\lambda)x = 0, \, x,y \in {\bf C}^n , \, x^*x=y^*y=1 , \, y^*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 $\Omega$ 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 $\, \mu \in \Omega \setminus W_q(P)$. 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.

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