Matrix Distributions

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

(RefLink[RandomVariate, paclet:ref/RandomVariate]) — generate random variate from matrix distributions

Matrix Distributions

(RefLink[MatrixNormalDistribution, paclet:ref/MatrixNormalDistribution]) — matrix-valued normal distribution

(RefLink[MatrixTDistribution, paclet:ref/MatrixTDistribution]) — matrix-valued Student t distribution

(RefLink[WishartMatrixDistribution, paclet:ref/WishartMatrixDistribution]) — matrix-valued chi-squared distribution

(RefLink[InverseWishartMatrixDistribution, paclet:ref/InverseWishartMatrixDistribution]) — matrix-valued inverse chi-squared distribution

Gaussian Structured Matrix Distributions

(RefLink[GaussianOrthogonalMatrixDistribution, paclet:ref/GaussianOrthogonalMatrixDistribution]) — symmetric matrices (GOE)

(RefLink[GaussianUnitaryMatrixDistribution, paclet:ref/GaussianUnitaryMatrixDistribution]) — Hermitian matrices (GUE)

(RefLink[GaussianSymplecticMatrixDistribution, paclet:ref/GaussianSymplecticMatrixDistribution]) — quaternionic Hermitian matrices (GSE)

Uniform Structured Matrix Distributions

(RefLink[CircularRealMatrixDistribution, paclet:ref/CircularRealMatrixDistribution]) — orthogonal matrices (CRE)

(RefLink[CircularUnitaryMatrixDistribution, paclet:ref/CircularUnitaryMatrixDistribution]) — unitary matrices (CUE)

(RefLink[CircularQuaternionMatrixDistribution, paclet:ref/CircularQuaternionMatrixDistribution]) — compact symplectic matrices (CQE)

(RefLink[CircularOrthogonalMatrixDistribution, paclet:ref/CircularOrthogonalMatrixDistribution]) — symmetric unitary matrices (COE)

(RefLink[CircularSymplecticMatrixDistribution, paclet:ref/CircularSymplecticMatrixDistribution]) — self-dual unitary matrices (CSE)

Special Limiting Matrix Property Distributions

(RefLink[WignerSemicircleDistribution, paclet:ref/WignerSemicircleDistribution]) — eigenvalues of random symmetric matrices

(RefLink[TracyWidomDistribution, paclet:ref/TracyWidomDistribution]) — scaled largest eigenvalue of random symmetric matrices

(RefLink[MarchenkoPasturDistribution, paclet:ref/MarchenkoPasturDistribution]) — singular values of Wishart matrices

General Matrix Property Distribution

(RefLink[MatrixPropertyDistribution, paclet:ref/MatrixPropertyDistribution]) — distribution properties of matrices

(RefLink[NExpectation, paclet:ref/NExpectation]) — compute expected properties of matrix distributions

(RefLink[NProbability, paclet:ref/NProbability]) — compute probabilities for properties of matrix distributions