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

Abstract

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.