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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the invariant faces associated with a cone-preserving map

Authors: Bit-Shun Tam and Hans Schneider
Journal: Trans. Amer. Math. Soc. 353 (2001), 209-245
MSC (2000): Primary 15A48; Secondary 47B65, 47A25, 46B42
Published electronically: July 12, 2000
MathSciNet review: 1707205
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Abstract: For an $n\!\times \! n$ nonnegative matrix $P$, an isomorphism is obtained between the lattice of initial subsets (of $\{ 1,\cdots ,n\}$) for $P$ and the lattice of $P$-invariant faces of the nonnegative orthant $\mathbb {R}^{n}_{+}$. Motivated by this isomorphism, we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone-preserving map on a polyhedral cone, formulated in terms of its invariant faces. In particular, we obtain the following extension of the famous Rothblum index theorem for a nonnegative matrix: If $A$ leaves invariant a polyhedral cone $K$, then for each distinguished eigenvalue $\lambda$ of $A$ for $K$, there is a chain of $m_\lambda$ distinct $A$-invariant join-irreducible faces of $K$, each containing in its relative interior a generalized eigenvector of $A$ corresponding to $\lambda$ (referred to as semi-distinguished $A$-invariant faces associated with $\lambda$), where $m_\lambda$ is the maximal order of distinguished generalized eigenvectors of $A$ corresponding to $\lambda$, but there is no such chain with more than $m_\lambda$ members. We introduce the important new concepts of semi-distinguished $A$-invariant faces, and of spectral pairs of faces associated with a cone-preserving map, and obtain several properties of a cone-preserving map that mostly involve these two concepts, when the underlying cone is polyhedral, perfect, or strictly convex and/or smooth, or is the cone of all real polynomials of degree not exceeding $n$ that are nonnegative on a closed interval. Plentiful illustrative examples are provided. Some open problems are posed at the end.

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Additional Information

Bit-Shun Tam
Affiliation: Department of Mathematics, Tamkang University, Tamsui, Taiwan 25137, ROC

Hans Schneider
Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706

Keywords: Cone-preserving map, nonnegative matrix, polyhedral cone, perfect cone, strictly convex smooth cone, spectral pair of a vector, spectral pair of a face, Perron-Schaefer condition, initial subset, semi-distinguished class, semi-distinguished invariant face, distinguished generalized eigenvector, chain of invariant faces
Received by editor(s): October 31, 1997
Received by editor(s) in revised form: March 11, 1999
Published electronically: July 12, 2000
Additional Notes: Research of the first author partially supported by the National Science Council of the Republic of China grant NSC 86-2115-M-032-002; the second author’s research partially supported by NSF grants DMS-9123318 and DMS-9424346.
Article copyright: © Copyright 2000 American Mathematical Society