Details

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

CircularRealMatrixDistribution[n], also referred to as the circular real ensemble (CRE), represents a statistical distribution over the orthogonal real matrices, namely real square matrices satisfying , where denotes the transpose of and denotes the identity matrix. Here, the parameter n is called the dimension parameter of the distribution and may be any positive integer.
Along with the circular quaternion matrix distribution (CircularQuaternionMatrixDistribution), the circular real matrix distribution is one of two major additions to the three original circle matrix ensembles (CircularOrthogonalMatrixDistribution, CircularSymplecticMatrixDistribution and CircularUnitaryMatrixDistribution) devised by Freeman Dyson in 1962. Probabilistically, the circular real matrix distribution represents a uniform distribution over the collection of orthogonal square matrices, while mathematically it is a so-called Haar measure on the orthogonal group . Matrix ensembles like the circular real 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 real 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,CircularRealMatrixDistribution[n]], written more concisely as ACircularRealMatrixDistribution[n], can be used to assert that a random matrix A is distributed according to a circular real 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 real matrix distribution may be computed using Tr, Eigenvalues and Norm, respectively. Such variates may also be examined with MatrixFunction and MatrixPower, while the entries of such variates can be plotted using MatrixPlot.
CircularRealMatrixDistribution is related to a number of other distributions. As discussed above, it is qualitatively similar to other circular matrix distributions such as CircularQuaternionMatrixDistribution, CircularOrthogonalMatrixDistribution, CircularSymplecticMatrixDistribution and CircularUnitaryMatrixDistribution. Originally, the circular matrix ensembles were derived as generalizations of the so-called Gaussian ensembles, and so CircularRealMatrixDistribution is related to GaussianOrthogonalMatrixDistribution, GaussianSymplecticMatrixDistribution and GaussianUnitaryMatrixDistribution. CircularRealMatrixDistribution is also related to MatrixNormalDistribution, MatrixTDistribution, WishartMatrixDistribution, InverseWishartMatrixDistribution, TracyWidomDistribution and WignerSemicircleDistribution.