On the largest eigenvalue of a symmetric nonnegative tensor

In this paper, some important spectral characterizations of symmetric nonnegative tensors are analyzed. In particular, it is shown that a symmetric nonnegative tensor has the following properties: (i) its spectral radius is zero if and only if it is a zero tensor; (ii) it is weakly irreducible (respectively, irreducible) if and only if it has a unique positive (respectively, nonnegative) eigenvalue-eigenvector; (iii) the minimax theorem is satisfied without requiring the weak irreducibility condition; and (iv) if it is weakly reducible, then it can be decomposed into some weakly irreducible tensors. In addition, the problem of finding the largest eigenvalue of a symmetric nonnegative tensor is shown to be equivalent to finding the global solution of a convex optimization problem. Subsequently, algorithmic aspects for computing the largest eigenvalue of symmetric nonnegative tensors are discussed.