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.
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This work was supported by NSFC Grant 11801100.
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Liang, C., Yang, Y. A note on Banach’s results concerning homogeneous polynomials associated with nonnegative tensors. Optim Lett 15, 419–429 (2021). https://doi.org/10.1007/s11590-020-01602-2
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DOI: https://doi.org/10.1007/s11590-020-01602-2