Generate a pseudorandom matrix:

Check that it is symmetric and positive definite:

Sample eigenvalues of a Wishart random matrix using MatrixPropertyDistribution:

Estimate joint distribution of eigenvalues:

Mean and variance:

Generate a single pseudorandom matrix:

Generate a set of pseudorandom matrices:

Sample at extended precision:

Compute statistical properties numerically:

Numerically approximate expectation of the largest matrix eigenvalue :

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare LogLikelihood for both distributions:

Skewness and kurtosis:

When n and p (the dimension of the covariance matrix Σ) are both large, the scaled largest eigenvalue of a matrix from a Wishart ensemble with identity covariance is approximately distributed as a Tracy–Widom distribution:

Sample the scaled largest eigenvalue:

Check goodness of fit with TracyWidomDistribution:

Algebraically independent components of a symmetric Wishart matrix have a known PDF:

Build the distribution of independent components of a Wishart matrix:

Find the joint distribution of a diagonal element:

Use MatrixPropertyDistribution to sample diagonal elements of Wishart matrices:

Check goodness of fit:

Use MatrixPropertyDistribution to represent the scaled eigenvalues of a Wishart random matrix with identity covariance:

The limiting distribution of eigenvalues follows MarchenkoPasturDistribution:

Compare the histogram of the eigenvalues with the PDF:

The expression , where and are, respectively, an independent Gaussian vector and Wishart matrix, follows HotellingTSquareDistribution:

Use MatrixPropertyDistribution to sample expressions :

Diagonal elements of a Wishart random matrix each follow a scaled χ² distribution:

Test against applicably scaled χ² distributions:

Diagonal elements are not independent:

For any nonzero vector and Wishart matrix with scale matrix , is χ² distributed: