Bapat, R. B. ; Olesky, D. D. ; van den Driessche, P.
(1995)
*Perron-Frobenius theory for a generalized Eigen problem*
Linear and Multilinear Algebra, 40
(2).
pp. 141-152.
ISSN 0308-1087

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Official URL: http://www.tandfonline.com/doi/abs/10.1080/0308108...

Related URL: http://dx.doi.org/10.1080/03081089508818429

## Abstract

Motivated by economic models, the generalized eigenvalue problem Ax=λ Bx is investigated under the conditions that A is nonnegative and irreducible, there is a nonnegative vector u such that Bu>Au, and b_{ij} ≤ _{ij} for all i#j. The last two conditions are equivalent to B-A being a nonsingular M-matrix. The focus is on generalizations of the Perron-Frobenius theory, the classical theory being recovered when B is the identity matrix. These generalizations include identification of a generalized eigenvalue ρ(A,B) in the interval (0,1) with a positive eigenvector, characterizations and easily computable bounds for ρ(A,B), and localization results for all generalized eigenvalues. Dropping the condition that A is irreducible, necessary and sufficient conditions for the problem to have a solution with x≥0 are formulated in terms of basic and final classes, which are natural extensions of these concepts in the classical theory.

Item Type: | Article |
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ID Code: | 77914 |

Deposited On: | 14 Jan 2012 15:41 |

Last Modified: | 14 Jan 2012 15:41 |

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