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On the numerical range map

Part of: Basic linear algebra

Published online by Cambridge University Press:  09 April 2009

M. Joswig  and
B. Straub
M. Joswig
Affiliation:
Fachbereich Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, D-10623 Berlin, Germany
B. Straub
Affiliation:
School of Mathematics, The University of New South Wales, Sydney, NSW 2052, Australia
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Abstract

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Let A ∈ ℒ(Cn) and A1, A2 be the unique Hermitian operators such that A = A1 + i A2. The paper is concerned with the differential structure of the numerical range map nA: x ↦ ((A1x, x), (A1x, x)) and its connection with certain natural subsets of the numerical range W(A) of A. We completely characterize the various sets of critical and regular points of the map nA as well as their respective images within W(A). In particular, we show that the plane algebraic curves introduced by R. Kippenhahn appear naturally in this context. They basically coincide with the image of the critical points of nA.

MSC classification

Secondary: 15A22: Matrix pencils 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 65 , Issue 2 , October 1998 , pp. 267 - 283
Copyright
Copyright © Australian Mathematical Society 1998

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