On matrices in finite free position
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912444704620544 |
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| author | Arizmendi, Octavio Lehner, Franz Rosenmann, Amnon |
| author_facet | Arizmendi, Octavio Lehner, Franz Rosenmann, Amnon |
| contents | We study pairs $(A,B)$ of square matrices that are in additive (resp. multiplicative) finite free position, that is, the characteristic polynomial $χ_{A+B}(x)$ (resp. $χ_{AB}(x)$) equals the additive finite free convolution $χ_{A}(x) \boxplus χ_{B}(x)$ (resp. the multiplicative finite free convolution $χ_{A}(x) \boxtimes χ_{B}(x)$), which equals the expected characteristic polynomial $\mathbb{E}_U [ χ_{A+U^* BU}(x) ]$ (resp. $\mathbb{E}_U [ χ_{AU^* BU}(x) ]$) over the set of unitary matrices $U$. We examine the lattice of (non-irreducible) affine algebraic sets of matrices consisting of finite free complementary pairs with respect to the additive (resp. multiplicative) convolution. We show that these pairs include the diagonal matrices vs. the principally balanced matrices, the upper (lower) triangular matrices vs. the upper (lower) triangular matrices with constant diagonal, and the scalar matrices vs. the set of all square matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_14343 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On matrices in finite free position Arizmendi, Octavio Lehner, Franz Rosenmann, Amnon Rings and Algebras Probability 60B20, 46L54 We study pairs $(A,B)$ of square matrices that are in additive (resp. multiplicative) finite free position, that is, the characteristic polynomial $χ_{A+B}(x)$ (resp. $χ_{AB}(x)$) equals the additive finite free convolution $χ_{A}(x) \boxplus χ_{B}(x)$ (resp. the multiplicative finite free convolution $χ_{A}(x) \boxtimes χ_{B}(x)$), which equals the expected characteristic polynomial $\mathbb{E}_U [ χ_{A+U^* BU}(x) ]$ (resp. $\mathbb{E}_U [ χ_{AU^* BU}(x) ]$) over the set of unitary matrices $U$. We examine the lattice of (non-irreducible) affine algebraic sets of matrices consisting of finite free complementary pairs with respect to the additive (resp. multiplicative) convolution. We show that these pairs include the diagonal matrices vs. the principally balanced matrices, the upper (lower) triangular matrices vs. the upper (lower) triangular matrices with constant diagonal, and the scalar matrices vs. the set of all square matrices. |
| title | On matrices in finite free position |
| topic | Rings and Algebras Probability 60B20, 46L54 |
| url | https://arxiv.org/abs/2309.14343 |