# CircularQuaternionMatrixDistribution

represents a circular quaternion matrix distribution with matrix dimensions {2 n,2 n} over the field of complex numbers.

# Details # Background & Context

• , also referred to as the circular quaternion ensemble (CQE), represents a statistical distribution over the unitary complex matrices, namely complex matrices of even dimension satisfying both and , where denotes the conjugate transpose of , the identity matrix, the transpose of and is a symplectic matrix of the form with the Kronecker product. The parameter n is called the dimension parameter of the distribution and may be any positive integer.
• Along with the circular real matrix ensemble (CircularRealMatrixDistribution), the circular quaternion matrix distribution is one of two major additions to the three original so-called circle matrix ensembles (CircularOrthogonalMatrixDistribution, CircularSymplecticMatrixDistribution and CircularUnitaryMatrixDistribution) devised by Freeman Dyson in 1962. Probabilistically, the circular quaternion matrix distribution represents a uniform distribution over the collection of compact symplectic square matrices, while mathematically it is a so-called Haar measure on the unitary symplectic group . Matrix ensembles like the circular quaternion 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 circular quaternion 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,CircularQuaternionMatrixDistribution[n]], written more concisely as ACircularQuaternionMatrixDistribution[n], can be used to assert that a random matrix A is distributed according to a circular quaternion matrix distribution. Such an assertion can then be used in functions such as MatrixPropertyDistribution.
• The trace, eigenvalues and norm of variates distributed according to circular quaternion matrix distribution may be computed using Tr, Eigenvalues and Norm, respectively. Such variates may also be examined with MatrixFunction, MatrixPower, and real quantities related thereto, such as the real part (Re), imaginary part (Im) and complex argument (Arg), can be plotted using MatrixPlot.
• CircularQuaternionMatrixDistribution is related to a number of other distributions. As discussed above, it is qualitatively similar to other circular matrix distributions such as CircularOrthogonalMatrixDistribution, CircularRealMatrixDistribution, CircularSymplecticMatrixDistribution and CircularUnitaryMatrixDistribution. Originally, the circular matrix ensembles were derived as generalizations of the so-called Gaussian ensembles, and so CircularQuaternionMatrixDistribution is related to GaussianOrthogonalMatrixDistribution, GaussianSymplecticMatrixDistribution and GaussianUnitaryMatrixDistribution. CircularQuaternionMatrixDistribution is also related to MatrixNormalDistribution, MatrixTDistribution, WishartMatrixDistribution, InverseWishartMatrixDistribution, TracyWidomDistribution and WignerSemicircleDistribution.

# Examples

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## Basic Examples(2)

Generate a pseudorandom CQE matrix:

It is unitary and preserves the symplectic matrix :

Represent the eigenvalues of a random matrix by MatrixPropertyDistribution and sample from it:

## Scope(3)

Generate a random matrix from unitary symplectic group :

Generate a set of random matrices from unitary symplectic group:

Compute statistical properties numerically:

## 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 2:

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

Compare the histogram with PDF of ChiSquareDistribution: