A note on Banach’s results concerning homogeneous polynomials associated with nonnegative tensors
| Autor: | Chang Liang, Yuning Yang |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
Polynomial 021103 operations research Control and Optimization 0211 other engineering and technologies 010103 numerical & computational mathematics 02 engineering and technology 01 natural sciences Singular value Multilinear form Order (group theory) Symmetric tensor Tensor 0101 mathematics Unit (ring theory) Eigenvalues and eigenvectors Mathematics |
| Zdroj: | Optimization Letters. 15:419-429 |
| ISSN: | 1862-4480 1862-4472 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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