The Perron generalized eigenspace and the spectral cone of a cone-preserving map

A unified treatment is offered to reprove known results on the following four highlights of the combinatorial spectral theory of nonnegative matrices, or to extend (or partly extend) the results to the setting of a linear map preserving a polyhedral proper (or proper) cone: the preferred-basis theor...

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Bibliographic Details
Published in:Linear algebra and its applications Vol. 393; pp. 375 - 429
Main Author: Tam, Bit-Shun
Format: Journal Article
Language:English
Published: Elsevier Inc 01.12.2004
Subjects:
Cone-preserving map
Height characteristic
Level characteristics
Majorization relation
Nonnegative matrix
Perron generalized eigenspace
Preferred-basis theorem
Principal component
Semi-distinguished invariant face
Spectral cone
Principal component
Nonnegative matrix
Height characteristic
Level characteristics
Semi-distinguished invariant face
Perron generalized eigenspace
Preferred-basis theorem
Spectral cone
15A48
Cone-preserving map
Majorization relation
ISSN:0024-3795, 1873-1856
Online Access:Get full text
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Summary:A unified treatment is offered to reprove known results on the following four highlights of the combinatorial spectral theory of nonnegative matrices, or to extend (or partly extend) the results to the setting of a linear map preserving a polyhedral proper (or proper) cone: the preferred-basis theorem, equivalent conditions for equality of the (graph-theoretic) level characteristic and the (spectral) height characteristic, the majorization relation between the two characteristics, and the relation between the combinatorial properties of a nonnegative matrix and the positivity of the individual entries in its principal components. This is achieved by employing the new concept of spectral cone of a cone-preserving map and combining the cone-theoretic methods developed in our previous papers on the geometric spectral theory of cone-preserving maps with the algebraic–analytic method introduced by Hartwig, Neumann and Rose and further exploited by Neumann and Schneider for nonnegative matrices.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2004.08.020