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On the existence of positive solutions of Au = λBu

Published online by Cambridge University Press:  20 January 2009

D. Kershaw
D. Kershaw
Affiliation:
Department of Mathematics, University of Lancaster
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It is well known that sufficient conditions for the existence of a positive vector u which satisfies the matrix equation Au = λu are that A should be non-negative and irreducible. This result, the qualitative part of the Perron-Frobenius theorem, has been proved in a variety of ways, one of the most attractive of which is that given by Alexandroff and Hopf in their treatise “ Topologie ”. The aim of this note is to show how their method can be adapted to deal with the generalised eigenvalue problem defined by Au = λBu where A and B are square matrices.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 18 , Issue 4 , December 1973 , pp. 281 - 285
Copyright
Copyright © Edinburgh Mathematical Society 1973

References

REFERENCE

(1) Mangasarian, O. L., Perron-Frobenius Properties of Ax-⋋Bx, J. Math. Anal. Appl. 36 (1971), 86–102.CrossRefGoogle Scholar