Matrix distributions are models for random matrices and have a variety of uses, including understanding linear algebra algorithms, direct modeling of quantum systems, and multivariate regression. Often key properties such as location of eigenvalues are of major interest.  The Wolfram Language provides efficient sampling of an extensive collection of matrix distributions, universal limiting distributions for key matrix properties, and a general framework and automation based on Monte Carlo simulation for studying any matrix property.

Simulation

RandomVariate — generate random variate from matrix distributions

Matrix Distributions

MatrixNormalDistribution — matrix-valued normal distribution

MatrixTDistribution — matrix-valued Student distribution

WishartMatrixDistribution — matrix-valued chi-squared distribution

InverseWishartMatrixDistribution — matrix‐valued inverse chi-squared distribution

Gaussian Structured Matrix Distributions

GaussianOrthogonalMatrixDistribution — symmetric matrices (GOE)

GaussianUnitaryMatrixDistribution — Hermitian matrices (GUE)

GaussianSymplecticMatrixDistribution — quaternionic Hermitian matrices (GSE)

Uniform Structured Matrix Distributions

CircularRealMatrixDistribution — orthogonal matrices (CRE)

CircularUnitaryMatrixDistribution — unitary matrices (CUE)

CircularQuaternionMatrixDistribution — compact symplectic matrices (CQE)

CircularOrthogonalMatrixDistribution — symmetric unitary matrices (COE)

CircularSymplecticMatrixDistribution — self-dual unitary matrices (CSE)

Special Limiting Matrix Property Distributions

WignerSemicircleDistribution — eigenvalues of random symmetric matrices

TracyWidomDistribution — scaled largest eigenvalue of random symmetric matrices

MarchenkoPasturDistribution — singular values of Wishart matrices

General Matrix Property Distribution

MatrixPropertyDistribution — distribution properties of matrices

NExpectation — compute expected properties of matrix distributions

NProbability — compute probabilities for properties of matrix distributions