# CircularRealMatrixDistribution

represents a circular real matrix distribution with matrix dimensions {n,n}.

# Examples

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## Basic Examples(2)

Generate a random CRE matrix:

Verify that the matrix is orthogonal:

Sample a random point on a sphere using MatrixPropertyDistribution:

The distribution of points over the sphere is uniform:

## Scope(3)

Generate a single random orthogonal matrix:

Generate a set of random orthogonal matrices:

Compute statistical properties numerically:

## Applications(2)

Sample EulerAngles of random special orthogonal matrices in 3D:

Check that the sample agrees with the expected distribution:

Visualize histograms of individual angles:

Sample points on by randomly rotating a fixed 4D vector:

Project the points to by Hopf map, for which the uniform measure on induces uniform measure on :

Project the points and bin them by the first coordinate of the projection:

Visualize the points on at different angles on :

## Properties & Relations(2)

Distribution of phase angle of the eigenvalues:

Compute the spacing between eigenvalues:

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

For eigenvectors of CircularRealMatrixDistribution with dimension large, the scaled modulus of the elements is distributed:

Compare the histogram with PDF of ChiSquareDistribution:

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

#### Text

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

#### CMS

Wolfram Language. 2015. "CircularRealMatrixDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CircularRealMatrixDistribution.html.

#### APA

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

#### BibTeX

@misc{reference.wolfram_2023_circularrealmatrixdistribution, author="Wolfram Research", title="{CircularRealMatrixDistribution}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/CircularRealMatrixDistribution.html}", note=[Accessed: 29-September-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2023_circularrealmatrixdistribution, organization={Wolfram Research}, title={CircularRealMatrixDistribution}, year={2015}, url={https://reference.wolfram.com/language/ref/CircularRealMatrixDistribution.html}, note=[Accessed: 29-September-2023 ]}