CircularRealMatrixDistribution
CircularRealMatrixDistribution[n]
represents a circular real matrix distribution with matrix dimensions {n,n}.
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 ACircularRealMatrixDistribution[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.
Examples
open all close allBasic Examples (2)
RandomVariate[CircularRealMatrixDistribution[3]]Verify that the matrix is orthogonal:
OrthogonalMatrixQ[%]Sample a random point on a sphere using MatrixPropertyDistribution:
RandomVariate[MatrixPropertyDistribution[r.{0, 0, 1}, rCircularRealMatrixDistribution[3]]]The distribution of points over the sphere is uniform:
Point[RandomVariate[MatrixPropertyDistribution[r.{0, 0, 1}, rCircularRealMatrixDistribution[3]], 10 ^ 3]]//Graphics3DScope (3)
Generate a single random orthogonal matrix:
RandomVariate[CircularRealMatrixDistribution[2]]Generate a set of random orthogonal matrices:
RandomVariate[CircularRealMatrixDistribution[2], 3]Compute statistical properties numerically:
det𝒟 = MatrixPropertyDistribution[Det[𝓂], 𝓂CircularRealMatrixDistribution[4]];NProbability[Δ == 1, Δdet𝒟]NProbability[Δ == -1, Δdet𝒟]Applications (2)
Sample EulerAngles of random special orthogonal matrices in 3D:
EulerAngles𝒟 = MatrixPropertyDistribution[EulerAngles[𝓂 / Det[𝓂]], 𝓂CircularRealMatrixDistribution[3]];angles = RandomVariate[EulerAngles𝒟, 10 ^ 4];Check that the sample agrees with the expected distribution:
DistributionFitTest[angles, ProductDistribution[UniformDistribution[{-Pi, Pi}], ProbabilityDistribution[(1/2)Sin[θ], {θ, 0, Pi}], UniformDistribution[{-Pi, Pi}]]]Visualize histograms of individual angles:
Histogram[#, Automatic, PDF]& /@ Transpose[angles]Sample points on 
by randomly rotating a fixed 4D vector:
p4d = RandomVariate[MatrixPropertyDistribution[r.UnitVector[4, 4], rCircularRealMatrixDistribution[4]], 5 10 ^ 4];AllTrue[p4d, RegionMember[Sphere[4]]]Project the points to 
by Hopf map, for which the uniform measure on 
induces uniform measure on 
:
hopf = {x, y, z, w} {(ArcTan[x, y] + Pi) / (2Pi), 2(x z + y w), 2(y z - x w), x ^ 2 + y ^ 2 - z ^ 2 - w ^ 2};Simplify[Norm[Rest[hopf[x, y, z, w]]], {x, y, z, w}∈Sphere[4]]Simplify[Det[Grad[hopf[x, y, z, w], {x, y, z, w}]], {x, y, z, w}∈Sphere[4]]Project the points and bin them by the first coordinate of the projection:
proj = hopf@@@p4d;
res = GatherBy[SortBy[proj, First], Quotient[First[#], 0.05]&];
res = Drop[res, None, None, 1];Visualize the points on 
at different angles on 
:
Animate[Graphics3D[{PointSize[Small], Point[res[[k]]]}, PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}], {{k, 1, "ϕ"}, 1, Length[res], 1}, TrackedSymbols :> k, SaveDefinitions -> True, AnimationRunning -> False]Properties & Relations (2)
Distribution of phase angle of the eigenvalues:
dim = 300;
args = RandomVariate[MatrixPropertyDistribution[Arg[Eigenvalues[x]], xCircularRealMatrixDistribution[dim]], 100];Histogram[Join@@args, {-Pi, Pi, 2Pi / 40}, PDF]Compute the spacing between eigenvalues:
diffs = Join@@Map[Composition[Differences, Sort], (args/2Pi)dim];Compare the histogram of sample level spacings with the closed form, also known as Wigner surmise for Dyson index 
:
WignerSurmisePDF[x_, β : 2] := 2(4 x / Pi) ^ 2 Exp[(-4 / Pi) x ^ 2]Show[Histogram[diffs, {0.1}, PDF], Plot[WignerSurmisePDF[x, 2], {x, 0, 2.5}]]For eigenvectors of CircularRealMatrixDistribution with dimension 
large, the scaled modulus of the elements is 
distributed:
𝒟 = MatrixPropertyDistribution[Flatten[Abs[Eigenvectors[𝓂]] ^ 2], 𝓂CircularRealMatrixDistribution[50]];data = 2Normalize[Join@@RandomVariate[𝒟, 100], Mean];Compare the histogram with PDF of ChiSquareDistribution:
Show[Histogram[data, "Scott", PDF], Plot[PDF[ChiSquareDistribution[2], x], {x, 0, 9}]]Text
Wolfram Research (2015), CircularRealMatrixDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/CircularRealMatrixDistribution.html.
CMS
Wolfram Language. 2015. "CircularRealMatrixDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CircularRealMatrixDistribution.html.
APA
Wolfram Language. (2015). CircularRealMatrixDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CircularRealMatrixDistribution.html