Peter Lancaster - Numerical ranges of selfadjoint quadratic matrix polynomials

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Peter Lancaster - Numerical ranges of selfadjoint quadratic matrix polynomials

PETER LANCASTER, Department of Mathematics and Statistics, University of Calgary Calgary, Alberta, T2N 1N4

Numerical ranges of selfadjoint quadratic matrix polynomials

Consider monic matrix polynomials $P(\lambda)= I\lambda^2 +A_1\lambda+A_0$ , where A₀ and A₁ are $n \times n$ Hermitian matrices and $\lambda$ is a complex variable. The numerical range of such a polynomial is

$\begin{displaymath}W(P) = \{\lambda \in \mathbb{C} : x^*P(\lambda)x = 0, \mbox{ for some nonzero } x \in \mathbb{C} ^n \} \end{displaymath}$

and it always contains the spectrum of P, $\sigma (P)$ , i.e. the set of zeros of $\det P(\lambda)$ . Properties of the numerical range are to be reviewed, taking advantage of the close connection between W(P)and the classical numerical range (field of values) of the (general) complex matrix A:=A₀+iA₁.

The study of eigenvalues and non-differentiable points on the boundary is of special interest. We consider also the problem of the numerical determination of W(P) and illustrate with examples generated with the help of ``matlab''. Comments are made on extension of the theory to more general polynomials $P(\lambda)$ , related factorization results, and the connection with the recently introduced ``quadratic numerical range'' when applied to the companion matrix of P.

This is a report on joint work with Panayotos Psarrakos.

Next: Jun Li - Asymptotic Up: Contributed Papers / Communications Previous: O. Kihel et C.