《16 Level characteristics corresponding to peripheral eigenvalues of a nonnegative matrix》.pdfVIP

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