Nonnegative Biquadratic Tensors
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913760653869056 |
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| author | Cui, Chunfeng Qi, Liqun |
| author_facet | Cui, Chunfeng Qi, Liqun |
| contents | An M-eigenvalue of a nonnegative biquadratic tensor is referred to as an M$^+$-eigenvalue if it has a pair of nonnegative M-eigenvectors. If furthermore that pair of M-eigenvectors is positive, then that M$^+$-eigenvalue is called an M$^{++}$-eigenvalue. A nonnegative biquadratic tensor has at least one M$^+$ eigenvalue, and the largest M$^+$-eigenvalue is both the largest M-eigenvalue and the M-spectral radius. For irreducible nonnegative biquadratic tensors, all the M$^+$-eigenvalues are M$^{++}$-eigenvalues. Although the M$^+$-eigenvalues of irreducible nonnegative biquadratic tensors are not unique in general, we establish a sufficient condition to ensure their uniqueness. For an irreducible nonnegative biquadratic tensor, the largest M$^+$-eigenvalue has a max-min characterization, while the smallest M$^+$-eigenvalue has a min-max characterization. A Collatz algorithm for computing the largest M$^+$-eigenvalues is proposed. Numerical results are reported. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16176 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonnegative Biquadratic Tensors Cui, Chunfeng Qi, Liqun Numerical Analysis An M-eigenvalue of a nonnegative biquadratic tensor is referred to as an M$^+$-eigenvalue if it has a pair of nonnegative M-eigenvectors. If furthermore that pair of M-eigenvectors is positive, then that M$^+$-eigenvalue is called an M$^{++}$-eigenvalue. A nonnegative biquadratic tensor has at least one M$^+$ eigenvalue, and the largest M$^+$-eigenvalue is both the largest M-eigenvalue and the M-spectral radius. For irreducible nonnegative biquadratic tensors, all the M$^+$-eigenvalues are M$^{++}$-eigenvalues. Although the M$^+$-eigenvalues of irreducible nonnegative biquadratic tensors are not unique in general, we establish a sufficient condition to ensure their uniqueness. For an irreducible nonnegative biquadratic tensor, the largest M$^+$-eigenvalue has a max-min characterization, while the smallest M$^+$-eigenvalue has a min-max characterization. A Collatz algorithm for computing the largest M$^+$-eigenvalues is proposed. Numerical results are reported. |
| title | Nonnegative Biquadratic Tensors |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2503.16176 |