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The main aim of this paper is to generalize matrix numerical ranges to the tensor case based on tensor norms. We show that the basic properties of matrix numerical ranges such as compactness and convexity are valid for tensor numerical ranges. We make use of convexity property to propose an algorithm for approximating tensor numerical ranges in which tensor eigenvalues are contained. Also we consider tensor numerical ranges based on inner products, however, they may not be convex in general. [ABSTRACT FROM AUTHOR]

Primitivity is an important concept in the spectral theory of nonnegative matrices and tensors. It is well-known that an irreducible matrix is primitive if and only if the greatest common divisor of all the cycles in the associated directed graph is equal to 1. The main aim of this paper is to establish a similar result, i.e., we show that a nonnegative tensor is primitive if and only if the greatest common divisor of all the cycles in the associated directed hypergraph is equal to 1. [ABSTRACT FROM AUTHOR]