free-matrix-laws

Utilities for distributions of matrix-valued non-commutative variables: Cauchy/resolvent transforms, operator-valued maps (e.g., \(\eta(B)=\sum_i A_i B A_i\)), and small fixed-point solvers.

Work-in-progress. Feedback welcome.

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Install

From GitHub (read-only users)

pip install -U "git+https://github.com/slavakargin/free-matrix-laws.git@main"

Iterating the operator-valued Cauchy transform

We solve the operator-valued semicircle equation $$ z\,G \;=\; I \;+\; \eta(G)\,G, \qquad \Im z>0, $$ where \(\eta(B)=\sum_{i=1}^s A_i\,B\,A_i^\ast\) is a completely positive (Kraus) map.

Fixed-point maps

The simplest iteration is $$ G \;\mapsto\; (\,zI - \eta(G)\,)^{-1}. $$

Following Helton–Rashidi Far–Speicher (IMRN 2007), a numerically friendlier choice is the half-averaged step $$ G \;\mapsto\; \tfrac12\Big[\,G \;+\; (\,zI - \eta(G)\,)^{-1}\Big], $$ which damps oscillations while preserving the correct fixed point.

Reference. J. W. Helton, R. Rashidi Far, R. Speicher,
Operator-valued Semicircular Elements: Solving a Quadratic Matrix Equation with Positivity Constraints, IMRN (2007).

In code: use free_matrix_laws.solve_cauchy_semicircle(z, A, ...), which implements the half-averaged step.