Computing the Noncommutative Inner Rank by Means of Operator-Valued Free Probability Theory

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, name...

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Bibliographic Details
Published in:Foundations of computational mathematics 2024-11
Main Authors: Hoffmann, Johannes, Mai, Tobias, Speicher, Roland
Format: Article
Language:English
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Summary: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.
ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-024-09684-5