Hostname: page-component-784d4fb959-k2xtr Total loading time: 0 Render date: 2025-07-17T00:38:44.171Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Information

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

References

[1]Bazer, J. and Yen, D. H. Y., ‘Lacunas of the Riemann matrix of symmetric-hyperbolic systems in two space variables’, Comm. Pure Appl. Math. 22 (1969), 279–333.CrossRefGoogle Scholar
[2]Binding, P. and Li, C-K., ‘Joint ranges of Hermitian matrices and simultaneous diagonalization’, Linear Algebra Appl. 151 (1991), 157–167.CrossRefGoogle Scholar
[3]Bochnak, J., Coste, M. and Roy, M.-F., Géométrie Algébrique Réelle (Springer, New York, 1987).Google Scholar
[4]Bredon, G. E., Topology and Geometry, Graduate Texts in Mathematics 139 (Springer, New York, 1993).CrossRefGoogle Scholar
[5]Engelking, R., General Topology, 2nd edition (Heldermann, Berlin, 1992).Google Scholar
[6]Fiedler, M., ‘Geometry of the numerical range of matrices’, Linear Algebra Appl. 37 (1981), 81–96.CrossRefGoogle Scholar
[7]Fiedler, M., ‘Numerical range of matrices and Levinger's theorem’, Linear Algebra Appl. 220 (1995), 171–180.CrossRefGoogle Scholar
[8]Hillman, J. A., Jefferies, B. R. F., Ricker, W. J. and Straub, B., ‘Differential properties of the numerical range map of pairs of matrics’, Linear Algebra Appl. 267 (1997), 317–334.CrossRefGoogle Scholar
[9]Kato, T., Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissensschaften 132 (Springer, Berlin, 1980).Google Scholar
[10]Kippenhahn, R., ‘Über den Wertevorrat einer Matrix’, Math. Nachr. 6 (1951), 193–228.CrossRefGoogle Scholar
[11]Lancaster, P., ‘On eigenvalues of matrices dependent on a parameter’, Number. Math. 6 (1964), 377–387.CrossRefGoogle Scholar
[12]Lancaster, P., Theory of Matrices (Academic Press, New York, 1969).Google Scholar
[13]Murnaghan, F. D., ‘On the field of values of a square matrix’, Proc. Nat. Acad. Sci. U.S.A. 18 (1932), 246–248.CrossRefGoogle ScholarPubMed
[14]Pease, M. C. III, Methods of Matrix Algebra (Academic Press, New York, 1965).Google Scholar
[15]Rellich, F., ‘Störungstheorie der Spektralzerlegung I’, Math. Ann. 113 (1937), 600–619.CrossRefGoogle Scholar
[16]Shafarevich, I. R., Basic Algebraic Geometry (Springer, New York, 1977).Google Scholar
[17]Spivak, M., A Comprehensive Introduction to Differential Geometry, volume V, 2nd edition (Publishor Perish, Inc., Wilmington, 1979).Google Scholar