《16 Level characteristics corresponding to peripheral eigenvalues of a nonnegative matrix》.pdfVIP
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《16 Level characteristics corresponding to peripheral eigenvalues of a nonnegative matrix》.pdf
Available online at
Linear Algebra and its Applications 429 (2008) 1719–1729
/locate/laa
Level characteristics corresponding to peripheral
eigenvalues of a nonnegative matrix
∗
Judith J. McDonald , DeAnne M. Morris
Box 643113, Department of Mathematics, Washington State University, Pullman, 99164-3113 WA, USA
Received 7 September 2006; accepted 18 May 2008
Available online 3 July 2008
Submitted by H. Schneider
Abstract
In this paper, we give necessary and sufficient conditions for a set of Jordan blocks to correspond to the
peripheral spectrum of a nonnegative matrix. For each eigenvalue, λ, the λ-level characteristic (with respect
to the spectral radius) is defined. The necessary and sufficient conditions include a requirement that the
λ-level characteristic is majorized by the λ-height characteristic. An algorithm which has been implemented
in MATLAB is given to determine when a multiset of Jordan blocks corresponds to the peripheral spectrum
of a nonnegative matrix. The algorithm is based on the necessary and sufficient conditions given in this
paper.
© 2008 Elsevier Inc. All rights reserved.
Keywords: Nonnegative matrix; Eventually nonnegative matrix; Peripheral spectrum; Jordan form; Level characteristic;
Height characteristic
1. Introduction
The Perron–Frobenius theorem is an important result which has prompted much research
pertaining to spectral properties of nonnegative matrices. Many combinatorial properties of the
spectrum of a nonnegative matrix, and its generalized eigenspace, are known (for example, see
[1,2,8]). Ma
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