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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2109.03360 |
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| _version_ | 1866913223317389312 |
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| author | Clark, Benjamin J. Paparella, Pietro |
| author_facet | Clark, Benjamin J. Paparella, Pietro |
| contents | In further pursuit of a solution to the celebrated nonnegative inverse eigenvalue problem, Loewy and London [Linear and Multilinear Algebra 6 (1978/79), no.~1, 83--90] posed the problem of characterizing all polynomials that preserve all nonnegative matrices of a fixed order. If $\mathscr{P}_n$ denotes the set of all polynomials that preserve all $n$-by-$n$ nonnegative matrices, then it is clear that polynomials with nonnegative coefficients belong to $\mathscr{P}_n$. However, it is known that $\mathscr{P}_n$ contains polynomials with negative entries. In this work, novel results for $\mathscr{P}_n$ with respect to the coefficients of the polynomials belonging to $\mathscr{P}_n$. Along the way, a generalization for the even-part and odd-part are given and shown to be equivalent to another construction that appeared in the literature. Implications for further research are discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_03360 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Polynomials that preserve nonnegative matrices Clark, Benjamin J. Paparella, Pietro Rings and Algebras In further pursuit of a solution to the celebrated nonnegative inverse eigenvalue problem, Loewy and London [Linear and Multilinear Algebra 6 (1978/79), no.~1, 83--90] posed the problem of characterizing all polynomials that preserve all nonnegative matrices of a fixed order. If $\mathscr{P}_n$ denotes the set of all polynomials that preserve all $n$-by-$n$ nonnegative matrices, then it is clear that polynomials with nonnegative coefficients belong to $\mathscr{P}_n$. However, it is known that $\mathscr{P}_n$ contains polynomials with negative entries. In this work, novel results for $\mathscr{P}_n$ with respect to the coefficients of the polynomials belonging to $\mathscr{P}_n$. Along the way, a generalization for the even-part and odd-part are given and shown to be equivalent to another construction that appeared in the literature. Implications for further research are discussed. |
| title | Polynomials that preserve nonnegative matrices |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2109.03360 |