represents a Gaussian orthogonal matrix distribution with matrix dimensions {n,n} and scale parameter σ.


represents a Gaussian orthogonal matrix distribution with unit scale parameter.


Background & Context


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Basic Examples  (4)

Generate a pseudorandom matrix from GOE:

Check that it is symmetric:

The entries of a matrix drawn from GaussianOrthogonalMatrixDistribution are jointly Gaussian and uncorrelated, with entries off the diagonal having half the variance of entries on the diagonal:

Use MatrixPropertyDistribution to sample eigenvalues of GOE matrices:

Mean and variance:

Scope  (4)

Generate a single pseudorandom matrix:

Generate a set of pseudorandom matrices:

Compute statistical properties numerically:

Estimate probability that the random matrix determinant is bounded away from zero:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare LogLikelihood of the distributions:

Applications  (3)

Sample eigenvalue spacing distribution in a 2×2 GOE matrix:

Compare the histogram with the closed form, also known as Wigner surmise for Dyson index :

Sample the joint distribution of eigenvalues of 2×2 GOE matrix:

Use RandomSample to randomly permute eigenvalues to compensate for algorithmspecific ordering:

Visualize estimated density:

Compare the estimated density to the known closed-form result:

Evaluate the density for the case of 2×2 GOE matrices:

Compare the density to the histogram density estimate from the sample:

Confirm the agreement with a goodness-of-fit test:

Illustrate complexity of matrix inversion using random symmetric matrices:

Properties & Relations  (4)

MatrixExp applied to with sampled from GaussianOrthogonalMatrixDistribution is symmetric and unitary:

Matrix elements of the upper-triangular part of a GOE matrix are independent Gaussian random variables:

Extract independent components of a 3×3 random matrix:

Use IndependenceTest to verify independence:

The spectral density of large GOE matrix converges to WignerSemicircleDistribution:

The distribution of the scaled largest eigenvalue of large GOE matrices converges to TracyWidomDistribution:

Wolfram Research (2015), GaussianOrthogonalMatrixDistribution, Wolfram Language function, (updated 2017).


Wolfram Research (2015), GaussianOrthogonalMatrixDistribution, Wolfram Language function, (updated 2017).


Wolfram Language. 2015. "GaussianOrthogonalMatrixDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017.


Wolfram Language. (2015). GaussianOrthogonalMatrixDistribution. Wolfram Language & System Documentation Center. Retrieved from


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@online{reference.wolfram_2024_gaussianorthogonalmatrixdistribution, organization={Wolfram Research}, title={GaussianOrthogonalMatrixDistribution}, year={2017}, url={}, note=[Accessed: 23-May-2024 ]}