Work

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

Year: 2024

Type: article

Abstract: 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.

Source: Foundations of Computational Mathematics

DOI: 10.1007/s10208-024-09684-5

Authors: Johannes Hoffmann, Tobias Mai, Roland Speicher

Institution: Saarland University

Language: English

Topic: Matrix Theory and Algorithms

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funder: Universität des Saarlandes