M-Tensors and Some Applications
We introduce M-tensors. This concept extends the concept of M-matrices. We denote Z-tensors as the tensors with nonpositive off-diagonal entries. We show that M-tensors must be Z-tensors and the maximal diagonal entry must be nonnegative. The diagonal elements of a symmetric M-tensor must be nonnega...
| Main Authors: | , , |
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| Format: | Journal Article |
| Published: |
Society for Industrial and Applied Mathematics
2014
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| Online Access: | http://hdl.handle.net/20.500.11937/35236 |
| _version_ | 1848754441838133248 |
|---|---|
| author | Zhang, L. Qi, L. Zhou, Guanglu |
| author_facet | Zhang, L. Qi, L. Zhou, Guanglu |
| author_sort | Zhang, L. |
| building | Curtin Institutional Repository |
| collection | Online Access |
| description | We introduce M-tensors. This concept extends the concept of M-matrices. We denote Z-tensors as the tensors with nonpositive off-diagonal entries. We show that M-tensors must be Z-tensors and the maximal diagonal entry must be nonnegative. The diagonal elements of a symmetric M-tensor must be nonnegative. A symmetric M-tensor is copositive. Based on the spectral theory of nonnegative tensors, we show that the minimal value of the real parts of all eigenvalues of an M-tensor is its smallest H+ -eigenvalue and also is its smallest H-eigenvalue. We show that a Z-tensor is an M-tensor if and only if all its H+ -eigenvalues are nonnegative. Some further spectral properties of M-tensors are given. We also introduce strong M-tensors, and some corresponding conclusions are given. In particular, we show that all H-eigenvalues of strong M-tensors are positive. We apply this property to study the positive definiteness of a class of multivariate forms associated with Z-tensors. We also propose an algorithm for testing the positive definiteness of such a multivariate form. |
| first_indexed | 2025-11-14T08:40:28Z |
| format | Journal Article |
| id | curtin-20.500.11937-35236 |
| institution | Curtin University Malaysia |
| institution_category | Local University |
| last_indexed | 2025-11-14T08:40:28Z |
| publishDate | 2014 |
| publisher | Society for Industrial and Applied Mathematics |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | curtin-20.500.11937-352362017-09-13T15:33:15Z M-Tensors and Some Applications Zhang, L. Qi, L. Zhou, Guanglu We introduce M-tensors. This concept extends the concept of M-matrices. We denote Z-tensors as the tensors with nonpositive off-diagonal entries. We show that M-tensors must be Z-tensors and the maximal diagonal entry must be nonnegative. The diagonal elements of a symmetric M-tensor must be nonnegative. A symmetric M-tensor is copositive. Based on the spectral theory of nonnegative tensors, we show that the minimal value of the real parts of all eigenvalues of an M-tensor is its smallest H+ -eigenvalue and also is its smallest H-eigenvalue. We show that a Z-tensor is an M-tensor if and only if all its H+ -eigenvalues are nonnegative. Some further spectral properties of M-tensors are given. We also introduce strong M-tensors, and some corresponding conclusions are given. In particular, we show that all H-eigenvalues of strong M-tensors are positive. We apply this property to study the positive definiteness of a class of multivariate forms associated with Z-tensors. We also propose an algorithm for testing the positive definiteness of such a multivariate form. 2014 Journal Article http://hdl.handle.net/20.500.11937/35236 10.1137/130915339 Society for Industrial and Applied Mathematics fulltext |
| spellingShingle | Zhang, L. Qi, L. Zhou, Guanglu M-Tensors and Some Applications |
| title | M-Tensors and Some Applications |
| title_full | M-Tensors and Some Applications |
| title_fullStr | M-Tensors and Some Applications |
| title_full_unstemmed | M-Tensors and Some Applications |
| title_short | M-Tensors and Some Applications |
| title_sort | m-tensors and some applications |
| url | http://hdl.handle.net/20.500.11937/35236 |