On the Best Rank-1 and Rank-(R1,R2,. . .,RN) Approximation of Higher-Order Tensors
SIAM Journal on Matrix Analysis and Applications
Convex Optimization
Finding the Largest Eigenvalue of a Nonnegative Tensor
SIAM Journal on Matrix Analysis and Applications
Eigenvalues of a real supersymmetric tensor
Journal of Symbolic Computation
An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor
Journal of Computational and Applied Mathematics
Further Results for Perron-Frobenius Theorem for Nonnegative Tensors
SIAM Journal on Matrix Analysis and Applications
Criterions for the positive definiteness of real supersymmetric tensors
Journal of Computational and Applied Mathematics
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For a nonnegative irreducible tensor, we give distribution properties of its eigenvalues. In particular, the spectral radius of a nonnegative irreducible tensor with positive trace is proved to be the unique eigenvalue on the spectral circle. Unlike the matrix setting, we give an example to present that this type of tensor is not always primitive. Thus, for a nonnegative irreducible tensor, the primitivity is a sufficient condition only for the spectral radius to be the unique eigenvalue on the spectral circle. Also, the stochastic tensor is defined, and we show that every nonnegative irreducible tensor with unit spectral radius is diagonally similar to a certain irreducible stochastic tensor. Based on this result, the minimax theorem for tensors is proved by using an alternative approach. Further, with the help of the minimax theorem, we illustrate that the problem of finding the spectral radius (largest singular value) of a nonnegative irreducible square (rectangular) tensor can be converted into a convex optimization problem. Additionally, we give an equivalent condition of irreducible nonnegative tensors. By this condition, one can easily determine whether or not a nonnegative tensor is irreducible.