A note on Banach’s results concerning homogeneous polynomials associated with nonnegative tensors

Abstract

A classic result of Banach states that the supreme of a multivariate homogenous polynomial is equivalent to that of its associated symmetric multilinear form over unit balls. Using the language of higher-order tensors in finite-dimensional spaces, this means that for a symmetric tensor, its largest singular value is in fact equivalent to the largest magnitude of its eigenvalues. This note strengthens Banach’s results on nonnegative higher-order tensors. It is shown that for a symmetric nonnegative irreducible tensor: (1) any singular value admitting a positive singular vector tuple must be an eigenvalue, where the singular vectors in the tuple must be equal to each other; i.e., they amount to an eigenvector; (2) when the order is odd, any nonnegative singular vector tuple corresponding to the largest singular value must also boil down to a positive eigenvector. These results give the flexibility of directly solving tensor eigenvalue problems via solving tensor singular value problems.