Sufficient conditions for total positivity, compounds, and Dodgson condensation
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914790088114176 |
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| author | Fallat, Shaun Gupta, Himanshu Johnson, Charles R. |
| author_facet | Fallat, Shaun Gupta, Himanshu Johnson, Charles R. |
| contents | A $n$-by-$n$ matrix is called totally positive ($TP$) if all its minors are positive and $TP_k$ if all of its $k$-by-$k$ submatrices are $TP$. For an arbitrary totally positive matrix or $TP_k$ matrix, we investigate if the $r$th compound ($1<r<n$) is in turn $TP$ or $TP_k$, and demonstrate a strong negative resolution in general. Focus is then shifted to Dodgson's algorithm for calculating the determinant of a generic matrix, and we analyze whether the associated condensed matrices are possibly totally positive or $TP_k$. We also show that all condensed matrices associated with a $TP$ Hankel matrix are $TP$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06069 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sufficient conditions for total positivity, compounds, and Dodgson condensation Fallat, Shaun Gupta, Himanshu Johnson, Charles R. Combinatorics Rings and Algebras 15B48 (Primary), 15A15, 15A24 (Secondary) A $n$-by-$n$ matrix is called totally positive ($TP$) if all its minors are positive and $TP_k$ if all of its $k$-by-$k$ submatrices are $TP$. For an arbitrary totally positive matrix or $TP_k$ matrix, we investigate if the $r$th compound ($1<r<n$) is in turn $TP$ or $TP_k$, and demonstrate a strong negative resolution in general. Focus is then shifted to Dodgson's algorithm for calculating the determinant of a generic matrix, and we analyze whether the associated condensed matrices are possibly totally positive or $TP_k$. We also show that all condensed matrices associated with a $TP$ Hankel matrix are $TP$. |
| title | Sufficient conditions for total positivity, compounds, and Dodgson condensation |
| topic | Combinatorics Rings and Algebras 15B48 (Primary), 15A15, 15A24 (Secondary) |
| url | https://arxiv.org/abs/2405.06069 |