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


Background & Context


open allclose all

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:

Introduced in 2015