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Title: A necessary and sufficient condition for existence of a positive perron vector
Authors: Hu, S
Qi, L 
Keywords: Nonnegative tensor
Perron-Frobenius theorem
Positive eigenvector
Spectral radius
Tensor eigenvalue
Issue Date: 2016
Publisher: Society for Industrial and Applied Mathematics
Source: SIAM journal on matrix analysis and applications, 2016, v. 37, no. 4, p. 1747-1770 How to cite?
Journal: SIAM journal on matrix analysis and applications 
Abstract: In 1907, Perron showed that a positive square matrix has a unique largest positive eigenvalue with a positive eigenvector. This result was extended to irreducible nonnegative matrices by Frobenius in 1912, and to irreducible nonnegative tensors and weakly irreducible nonnegative tensors recently. This result is a fundamental result in matrix theory and has found wide applications in probability theory, internet search engines, spectral graph and hypergraph theory, etc. In this paper, we give a necessary and sufficient condition for the existence of such a positive eigenvector, i.e., a positive Perron vector, for a nonnegative tensor. We show that every nonnegative tensor has a canonical nonnegative partition form, from which we introduce strongly nonnegative tensors. A tensor is called strongly nonnegative if the spectral radius of each genuine weakly irreducible block is equal to the spectral radius of the tensor, which is strictly larger than the spectral radius of any other block. We prove that a nonnegative tensor has a positive Perron vector if and only if it is strongly nonnegative. The proof is nontrivial. Numerical results for finding a positive Perron vector are reported.
ISSN: 0895-4798
EISSN: 1095-7162
DOI: 10.1137/15M1051828
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