Annals of Functional Analysis

A max version of Perron--Frobenius theorem for nonnegative tensor

Hamid Reza Afshin and Ali Reza Shojaeifard

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In this paper we generalize the max algebra system of nonnegative matrices to the class of nonnegative tensors and derive its fundamental properties. If $\mathbb{A} \in \Re_ + ^{\left[ {m,n} \right]}$ is a nonnegative essentially positive tensor such that satisfies the condition class NC, we prove that there exist $\mu \left( \mathbb{A} \right)$ and a corresponding positive vector $x$ such that $\mathop {\max }\limits_{1 \le{i_2}\cdots {i_m} \le n} \left\{ {{a_{i{i_2}\cdots {i_m}}}{x_{{i_2}}}\cdots {x_{{i_m}}}} \right\}=\mu \left( \mathbb{A} \right) x_i^{m - 1},\,\,\,\,i = 1,2,\cdots ,n.$ This theorem, is well known as the max algebra version of Perron--Frobenius theorem for this new system.

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Ann. Funct. Anal., Volume 6, Number 3 (2015), 145-154.

First available in Project Euclid: 17 April 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A18: Eigenvalues, singular values, and eigenvectors
Secondary: 15A69: Multilinear algebra, tensor products 74B99: None of the above, but in this section

Perron--Frobenius theory max algebra nonnegative tensor


Afshin, Hamid Reza; Shojaeifard, Ali Reza. A max version of Perron--Frobenius theorem for nonnegative tensor. Ann. Funct. Anal. 6 (2015), no. 3, 145--154. doi:10.15352/afa/06-3-12.

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