Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T03:20:47.005Z Has data issue: false hasContentIssue false
  • English
  • Français

Inverse multi-parameter eigenvalue problems for matrices*

Published online by Cambridge University Press:  14 November 2011

Patrick J. Browne  and
B. D. Sleeman
Patrick J. Browne
Affiliation:
Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, Canada T2N 1N4
B. D. Sleeman
Affiliation:
Department of Mathematical Sciences, University of Dundee, Dundee DD1 4HN, Scotland
Get access Rights & Permissions [Opens in a new window]

Synopsis

We study the possibility of perturbing a matrix A by a diagonal matrix so that an eigenvalue problem with leading matrix A has specifiedeigenvalues when A is replaced by A+D. The particular cases presented are the one-parameter generalized eigenvalue problem (A× = λB μ×, a two-parameter eigenvalue problem (A + λB + μC)× = 0, a linked system ofsuch two-parameter problems and a quadratic eigenvalue problem (A + λB + λ2C)× = 0. The work extends results of Hadeler for the classical problem A× = λ×.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 100 , Issue 1-2 , 1985 , pp. 29 - 38
Copyright
Copyright © Royal Society of Edinburgh 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Biegler-König, F. W.. Sufficient conditions for the solubility of inverse eigenvalue problems. Linear Algebra Appl. 40 (1981), 89–100.CrossRefGoogle Scholar
2Binding, P. and Browne, P. J.. Spectral properties of two-parameter eigenvalue problems. Proc. Roy. Soc. Edinburgh Sect. A 89 (1981), 157–173.CrossRefGoogle Scholar
3Binding, P. and Browne, P. J.. Comparison cones for multiparameter eigenvalue problems. J. Math. Anal. Appl 77 (1980), 132–149.CrossRefGoogle Scholar
4Browne, P. J.. Multiparameter problems: the last decade. Proceedings 1982 Dundee Conference on Ordinary and Partial Differential Equations. Lecture Notes in Mathematics 964, 95–109 (Berlin: Springer: 1982).Google Scholar
5Friedland, S. and Leibl, P.. On an inverse problem for non-negative and eventually non-negative matrices. Israel J. Math 29 (1978), 43–60.CrossRefGoogle Scholar
6Hadeler, K. P.. Ein inverses Eigenwert problem. Linear Algebra Appl 1 (1968), 83–101.CrossRefGoogle Scholar
7Hadeler, K. P.. Multiplikative inverse Eigenwertprobleme. Linear Algebra Appl 2 (1969), 65–86.CrossRefGoogle Scholar
8Hadeler, K. P.. Existenz- und Eindeutigkeitssätze für inverse Eigenwertaufgaben mit Hilfe des topologischen Abbildungsgrades. Arch. Rational Mech. Anal. 42 (1971), 317–322.CrossRefGoogle Scholar
9Oliveira, G. N.. Matrices with prescribed principal elements and singular values. Canad. Math. Bull. 14 (1971), 247–249.CrossRefGoogle Scholar
10Oliveira, G. N.. On the multiplicative inverse eigenvalue problem. Canad. Math. Bull 15 (1972), 189–193.CrossRefGoogle Scholar