Primitivity, the Convergence of the NQZ Method, and the Largest Eigenvalue for Nonnegative Tensors

Abstract

We define and study (nonnegative) primitive tensors. Many important characterizations of primitive matrices can be extended to nonnegative tensors. The NQZ method for calculating the largest eigenvalue of an irreducible tensor is proposed in [8] and is an extension of Collatz’s method. In particular, the convergence of the NQZ method for finding the largest eigenvalue of a nonnegative irreducible tensor, proposed by Ng, Qi, and Zhou, is proved under the primitive assumption. This fact can then be used to find the largest eigenvalue of any nonnegative irreducible tensor.