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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the invariant faces associated with a cone-preserving map
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by Bit-Shun Tam and Hans Schneider PDF
Trans. Amer. Math. Soc. 353 (2001), 209-245 Request permission


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
  • Email:
  • Hans Schneider
  • Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706
  • Email:
  • 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.
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 353 (2001), 209-245
  • MSC (2000): Primary 15A48; Secondary 47B65, 47A25, 46B42
  • DOI:
  • MathSciNet review: 1707205