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.

中文翻译：

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通过算子值自由概率论计算非交换内秩

我们解决了 Edmonds 问题的非交换版本，它要求确定矩阵在非交换变量中的内部秩。我们提供了一种算法来计算这个内部秩，方法是将问题与自由概率论中基本对象的分布联系起来，即算子值半圆元素。我们必须求解一个矩阵值二次方程，为此，我们对求解方程的定点算法提供精确的分析和数控。数值示例显示了算法的效率。