# GaussianOrthogonalMatrixDistribution

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.

# Examples

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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, https://reference.wolfram.com/language/ref/GaussianOrthogonalMatrixDistribution.html (updated 2017).

#### Text

Wolfram Research (2015), GaussianOrthogonalMatrixDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/GaussianOrthogonalMatrixDistribution.html (updated 2017).

#### CMS

Wolfram Language. 2015. "GaussianOrthogonalMatrixDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/GaussianOrthogonalMatrixDistribution.html.

#### APA

Wolfram Language. (2015). GaussianOrthogonalMatrixDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GaussianOrthogonalMatrixDistribution.html

#### BibTeX

@misc{reference.wolfram_2023_gaussianorthogonalmatrixdistribution, author="Wolfram Research", title="{GaussianOrthogonalMatrixDistribution}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/GaussianOrthogonalMatrixDistribution.html}", note=[Accessed: 27-September-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2023_gaussianorthogonalmatrixdistribution, organization={Wolfram Research}, title={GaussianOrthogonalMatrixDistribution}, year={2017}, url={https://reference.wolfram.com/language/ref/GaussianOrthogonalMatrixDistribution.html}, note=[Accessed: 27-September-2023 ]}