# 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.

# Details • GaussianOrthogonalMatrixDistribution is also known as Gaussian orthogonal ensemble, or GOE.
• GaussianOrthogonalMatrixDistribution is a distribution of a symmetric matrix , where is a square matrix with independent identically distributed matrix elements that follow NormalDistribution[0,σ].
• The probability density for matrix is proportional to .
• The scale parameter σ can be any positive number, and n can be any positive integer.
• GaussianOrthogonalMatrixDistribution can be used with such functions as MatrixPropertyDistribution, EstimatedDistribution, and RandomVariate.

# Background & Context

• , also referred to as a Gaussian orthogonal ensemble (GOE), represents a statistical distribution over the real symmetric matrices, namely square real matrices that satisfy , where denotes the transpose of . Matrices distributed according to GaussianOrthogonalMatrixDistribution have probability densities proportional to . Furthermore, the collection of all entries is an independent collection of real variates distributed identically according to NormalDistribution[0,σ]. is parameterized by a positive integer n (the dimension parameter) and a positive real number σ (the scale parameter). Despite the name "Gaussian orthogonal matrix distribution", matrices belonging to this distribution need not be orthogonal.
• The one-parameter form is equivalent to .
• Along with the Gaussian symplectic and Gaussian unitary distributions (GaussianSymplecticMatrixDistribution and GaussianUnitaryMatrixDistribution, respectively), the Gaussian orthogonal matrix distribution was one of three Gaussian matrix ensembles originally suggested by Eugene Wigner as a tool to study fluctuations in nuclear physics. Mathematically, the GOE is invariant under conjugation by orthogonal matrices, while physically modeling Hamiltonians with time-reversal symmetry. Matrix ensembles like the Gaussian orthogonal matrix distribution are of considerable importance in the study of random matrix theory, as well as in various branches of physics and mathematics.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Gaussian orthogonal matrix distribution, and the mean, median, variance, raw moments and central moments of a collection of such variates may then be computed using Mean, Median, Variance, Moment and CentralMoment, respectively. Distributed[A,GaussianOrthogonalMatrixDistribution[σ,n]], written more concisely as AGaussianOrthogonalMatrixDistribution[σ,n], can be used to assert that a random matrix A is distributed according to a Gaussian orthogonal matrix distribution. Such an assertion can then be used in functions such as MatrixPropertyDistribution.
• The trace, eigenvalues and norm of variates distributed according to Gaussian orthogonal matrix distribution may be computed using Tr, Eigenvalues and Norm, respectively. Such variates may also be examined with MatrixFunction and MatrixPower, while the entries of such variates can be plotted using MatrixPlot.
• GaussianOrthogonalMatrixDistribution is related to a number of other distributions. As discussed above, it is qualitatively similar to other Gaussian matrix distributions GaussianSymplecticMatrixDistribution and GaussianUnitaryMatrixDistribution. Generalizations of the Gaussian ensembles include the so-called circular matrix ensembles, and so GaussianOrthogonalMatrixDistribution is also related to CircularOrthogonalMatrixDistribution, CircularQuaternionMatrixDistribution, CircularRealMatrixDistribution, CircularSymplecticMatrixDistribution and CircularUnitaryMatrixDistribution. GaussianOrthogonalMatrixDistribution is also related to MatrixNormalDistribution, MatrixTDistribution, WishartMatrixDistribution, InverseWishartMatrixDistribution, TracyWidomDistribution and WignerSemicircleDistribution.

# 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:

Introduced in 2015
(10.3)
|
Updated in 2017
(11.1)