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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2015
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1510.02515 |
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| _version_ | 1866918161231642624 |
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| author | Huang, Huajun Oeding, Luke |
| author_facet | Huang, Huajun Oeding, Luke |
| contents | We solve the Symmetrized Principal Minor Assignment Problem, that is we show how to determine if for a given vector $v\in \mathbb{C}^{n}$ there is an $n\times n$ matrix that has all $i\times i$ principal minors equal to $v_{i}$. We use a special isomorphism (a non-linear change of coordinates to cycle-sums) that simplifies computation and reveals hidden structure. We use the symmetries that preserve symmetrized principal minors and cycle-sums to treat 3 cases: symmetric, skew-symmetric and general square matrices. We describe the matrices that have such symmetrized principal minors as well as the ideal of relations among symmetrized principal minors / cycle-sums. We also connect the resulting algebraic varieties of symmetrized principal minors to tangential and secant varieties, and Eulerian polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1510_02515 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Symmetrization of Principal Minors and Cycle-Sums Huang, Huajun Oeding, Luke Algebraic Geometry 05A05, 15A69, 15B05 We solve the Symmetrized Principal Minor Assignment Problem, that is we show how to determine if for a given vector $v\in \mathbb{C}^{n}$ there is an $n\times n$ matrix that has all $i\times i$ principal minors equal to $v_{i}$. We use a special isomorphism (a non-linear change of coordinates to cycle-sums) that simplifies computation and reveals hidden structure. We use the symmetries that preserve symmetrized principal minors and cycle-sums to treat 3 cases: symmetric, skew-symmetric and general square matrices. We describe the matrices that have such symmetrized principal minors as well as the ideal of relations among symmetrized principal minors / cycle-sums. We also connect the resulting algebraic varieties of symmetrized principal minors to tangential and secant varieties, and Eulerian polynomials. |
| title | Symmetrization of Principal Minors and Cycle-Sums |
| topic | Algebraic Geometry 05A05, 15A69, 15B05 |
| url | https://arxiv.org/abs/1510.02515 |