Orthogonal eigenvectors and singular vectors of tensors
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912446036312064 |
|---|---|
| author | Ribot, Alvaro Seigal, Anna Zwiernik, Piotr |
| author_facet | Ribot, Alvaro Seigal, Anna Zwiernik, Piotr |
| contents | The spectral theorem says that a real symmetric matrix has an orthogonal basis of eigenvectors and that, for a matrix with distinct eigenvalues, the basis is unique (up to signs). In this paper, we study the symmetric tensors with an orthogonal basis of eigenvectors and show that, for a generic such tensor, the orthogonal basis is unique. This resolves a conjecture by Mesters and Zwiernik. We also study the non-symmetric setting. The singular value decomposition says that a real matrix has an orthogonal basis of singular vector pairs and that, for a matrix with distinct singular values, the basis is unique (up to signs). We describe the tensors with an orthogonal basis of singular vectors and show that a generic such tensor has a unique orthogonal basis, with one exceptional format: order four binary tensors. We use these results to propose a new tensor decomposition that generalizes an orthogonally decomposable decomposition and specializes the Tucker decomposition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19009 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Orthogonal eigenvectors and singular vectors of tensors Ribot, Alvaro Seigal, Anna Zwiernik, Piotr Spectral Theory Algebraic Geometry 15A18, 15A69, 13P15 The spectral theorem says that a real symmetric matrix has an orthogonal basis of eigenvectors and that, for a matrix with distinct eigenvalues, the basis is unique (up to signs). In this paper, we study the symmetric tensors with an orthogonal basis of eigenvectors and show that, for a generic such tensor, the orthogonal basis is unique. This resolves a conjecture by Mesters and Zwiernik. We also study the non-symmetric setting. The singular value decomposition says that a real matrix has an orthogonal basis of singular vector pairs and that, for a matrix with distinct singular values, the basis is unique (up to signs). We describe the tensors with an orthogonal basis of singular vectors and show that a generic such tensor has a unique orthogonal basis, with one exceptional format: order four binary tensors. We use these results to propose a new tensor decomposition that generalizes an orthogonally decomposable decomposition and specializes the Tucker decomposition. |
| title | Orthogonal eigenvectors and singular vectors of tensors |
| topic | Spectral Theory Algebraic Geometry 15A18, 15A69, 13P15 |
| url | https://arxiv.org/abs/2506.19009 |