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

TL;DR

This paper introduces an algorithm to compute the inner rank of matrices with noncommuting variables by leveraging operator-valued free probability theory, specifically using operator-valued semicircular elements and solving matrix quadratic equations.

Contribution

It presents a novel method connecting noncommutative algebra with free probability, providing an efficient algorithm for inner rank calculation.

Findings

Algorithm effectively computes noncommutative inner rank.

Numerical examples demonstrate the algorithm's efficiency.

Provides analytical and numerical control over the fixed point method.

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.