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


  • CircularRealMatrixDistribution is also known as circular real ensemble, or CRE.
  • CircularRealMatrixDistribution represents a uniform distribution over the orthogonal square matrices of dimension n, also known as the Haar measure on the orthogonal group .
  • The dimension parameter n can be any positive integer.
  • CircularRealMatrixDistribution can be used with such functions as MatrixPropertyDistribution and RandomVariate.

Background & Context


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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,


Wolfram Research (2015), CircularRealMatrixDistribution, Wolfram Language function,


Wolfram Language. 2015. "CircularRealMatrixDistribution." Wolfram Language & System Documentation Center. Wolfram Research.


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


@misc{reference.wolfram_2024_circularrealmatrixdistribution, author="Wolfram Research", title="{CircularRealMatrixDistribution}", year="2015", howpublished="\url{}", note=[Accessed: 14-June-2024 ]}


@online{reference.wolfram_2024_circularrealmatrixdistribution, organization={Wolfram Research}, title={CircularRealMatrixDistribution}, year={2015}, url={}, note=[Accessed: 14-June-2024 ]}