Computing the noncommutative inner rank by means of operator-valued free probability theory
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912148446248960 |
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| author | Hoffmann, Johannes Mai, Tobias Speicher, Roland |
| author_facet | Hoffmann, Johannes Mai, Tobias Speicher, Roland |
| contents | We address the noncommutative version of the Edmonds' problem, which asks to determine the inner rank of a matrix in noncommuting variables. We provide an algorithm for the calculation of this inner rank by relating the problem with the distribution of a basic object in free probability theory, namely operator-valued semicircular elements. We have to solve a matrix-valued quadratic equation, for which we provide precise analytical and numerical control on the fixed point algorithm for solving the equation. Numerical examples show the efficiency of the algorithm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_03667 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Computing the noncommutative inner rank by means of operator-valued free probability theory Hoffmann, Johannes Mai, Tobias Speicher, Roland Operator Algebras Rings and Algebras 46L54, 65J15, 12E15 We address the noncommutative version of the Edmonds' problem, which asks to determine the inner rank of a matrix in noncommuting variables. We provide an algorithm for the calculation of this inner rank by relating the problem with the distribution of a basic object in free probability theory, namely operator-valued semicircular elements. We have to solve a matrix-valued quadratic equation, for which we provide precise analytical and numerical control on the fixed point algorithm for solving the equation. Numerical examples show the efficiency of the algorithm. |
| title | Computing the noncommutative inner rank by means of operator-valued free probability theory |
| topic | Operator Algebras Rings and Algebras 46L54, 65J15, 12E15 |
| url | https://arxiv.org/abs/2308.03667 |