Computing the noncommutative inner rank by means of operator-valued free probability theory

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,...

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Veröffentlicht in:arXiv.org 2024-06
Hauptverfasser: Hoffmann, Johannes, Mai, Tobias, Speicher, Roland
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Algorithms
Mathematics - Operator Algebras
Mathematics - Rings and Algebras
Numerical controls
Probability theory
Quadratic equations
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description 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.
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subjects Algorithms
Mathematics - Operator Algebras
Mathematics - Rings and Algebras
Numerical controls
Probability theory
Quadratic equations
title Computing the noncommutative inner rank by means of operator-valued free probability theory
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