Generate a pseudorandom COE matrix:

It is both unitary and symmetric:

Sample a random point on a sphere using MatrixPropertyDistribution:

The distribution is visibly clustered around the axes:

Generate a single pseudorandom matrix:

Generate a set of pseudorandom matrices:

Compute statistical properties numerically:

Define distribution of complex arguments of random matrix eigenvalues:

Sample the phases of eigenvalues followed by random permutations:

Visualize joint phase distribution together with the closed-form PDF:

The joint distribution of the eigenvalues for CircularOrthogonalMatrixDistribution is also Boltzmann distribution of Dyson's Coulomb gas on a circle with inverse temperature . The average Hamiltonian per particle of the system is (without kinetic terms):

Define the distribution of the value of Hamiltonian on random COE matrix:

Compute the sample mean of the Hamiltonian for systems of different size:

Plot the sample means and compare them with thermodynamic limit:

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 CircularOrthogonalMatrixDistribution with dimension large, the scaled modulus of the elements is distributed:

Compare the histogram with PDF of ChiSquareDistribution:

A matrix from CircularOrthogonalMatrixDistribution need not be orthogonal:

Use CircularRealMatrixDistribution to sample a random orthogonal real-valued matrix:

Matrix from CircularOrthogonalMatrixDistribution can be represented as matrix , where matrix follows CircularUnitaryMatrixDistribution: