# CircularSymplecticMatrixDistribution

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

# Details # Background & Context

• , also referred to as the circular symplectic ensemble (CSE), represents a statistical distribution over the unitary and self-dual complex matrices, namely complex square 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. Despite the name "circular symplectic matrix distribution", matrices belonging to this distribution need not be symplectic.
• Along with the circular orthogonal and circular unitary matrix distributions (CircularOrthogonalMatrixDistribution and CircularUnitaryMatrixDistribution, respectively), the circular symplectic matrix distribution was one of three so-called circle matrix ensembles originally devised by Freeman Dyson in 1962 as a tool to study quantum mechanics. Probabilistically, the circular symplectic matrix distribution represents a uniform distribution over the self-dual unitary quaternionic square matrices. Matrix ensembles like the circular symplectic 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 symplectic 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,CircularSymplecticMatrixDistribution[n]], written more concisely as ACircularSymplecticMatrixDistribution[n] , can be used to assert that a random matrix A is distributed according to a circular symplectic 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 symplectic 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.
• CircularSymplecticMatrixDistribution is related to a number of other distributions. As discussed above, it is qualitatively similar to other circular matrix distributions such as CircularQuaternionMatrixDistribution, CircularRealMatrixDistribution, CircularOrthogonalMatrixDistribution and CircularUnitaryMatrixDistribution. Originally, the circular matrix ensembles were derived as generalizations of the so-called Gaussian ensembles, and so CircularSymplecticMatrixDistribution is related to GaussianOrthogonalMatrixDistribution, GaussianSymplecticMatrixDistribution and GaussianUnitaryMatrixDistribution. CircularSymplecticMatrixDistribution is also related to MatrixNormalDistribution, MatrixTDistribution, WishartMatrixDistribution, InverseWishartMatrixDistribution, TracyWidomDistribution and WignerSemicircleDistribution.

# Examples

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## Basic Examples(2)

Generate a pseudorandom matrix from unitary symplectic group:

 In:= The random matrix is unitary:

 In:= Out= It also verifies the symplectic self-duality condition:

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

 In:= Out= ## Possible Issues(1)

Introduced in 2015
(10.3)