MatrixPropertyDistribution
MatrixPropertyDistribution[expr,xmdist]
represents the distribution of the matrix property expr where the matrix-valued random variable x follows the matrix distribution mdist.
MatrixPropertyDistribution[expr,{x1mdist1,x2mdist2,…}]
represents the distribution where x1, x2, … are independent and follow the matrix distributions mdist1, mdist2, ….
Details
- MatrixPropertyDistribution is a transformation from the space of matrices to some property that is often of much lower dimension.
- MatrixPropertyDistribution is typically used to study properties of a distribution of matrices such as eigenvalues, singular values, determinant, norm or any property that can be computed.
- xdist can be entered as x
dist
dist or x∖[Distributed]dist. - MatrixPropertyDistribution can be used with such functions as NProbability, NExpectation, and RandomVariate.
Examples
open all close allBasic Examples (3)
Approximate the mean of 
for Gaussian orthogonal matrix 
:
Mean[MatrixPropertyDistribution[Tr[x.x], xGaussianOrthogonalMatrixDistribution[2]]]//NDraw a sample solution of a random linear system:
RandomVariate[MatrixPropertyDistribution[LinearSolve[m, {v1, v2, v3}], {mCircularRealMatrixDistribution[3], {v1, v2, v3}MultinormalDistribution[{0, 0, 0}, IdentityMatrix[3]]}]]Estimate distribution of Log10 of condition number of a random matrix:
matrixCondNum[mat_] := Norm[Inverse[mat]]Norm[mat]log10cn = RandomVariate[MatrixPropertyDistribution[Log10[matrixCondNum[x]], xWishartMatrixDistribution[10, HilbertMatrix[5]]], 10 ^ 5];SmoothHistogram[log10cn, Filling -> Axis]Scope (3)
Define distribution of a scalar-valued function of matrix argument:
MatrixPropertyDistribution[Tr[𝓂.𝓂^], 𝓂MatrixNormalDistribution[IdentityMatrix[3], IdentityMatrix[3]]]Approximate the mean of the function:
Mean[%]//NDefine distribution of a vector-valued function of matrix argument:
MatrixPropertyDistribution[Diagonal[𝓂], 𝓂InverseWishartMatrixDistribution[5, IdentityMatrix[3]]]RandomVariate[%, 3]Define distribution from a random matrix and a random vector:
cov = {{1, 1 / 2, 0}, {1 / 2, 1, 1 / 2}, {0, 1 / 2, 1}};𝒟 = MatrixPropertyDistribution[{x1, x2, x3}.𝓂.{x1, x2, x3}, {𝓂WishartMatrixDistribution[100, cov], {x1, x2, x3}MultinormalDistribution[{0, 0, 0}, cov]}];Approximate quartiles of the distribution:
RandomVariate[𝒟, 10 ^ 5]//QuartilesApplications (4)
Sample determinant of matrix from GaussianOrthogonalMatrixDistribution:
n = 2;
dets = RandomVariate[MatrixPropertyDistribution[Det[x], xGaussianOrthogonalMatrixDistribution[n]], 10 ^ 6];Compare the histogram to the known PDF:
detpdf[y_] := (1/Sqrt[2])Exp[y] Piecewise[{{Erfc[Sqrt[2 y]], y ≥ 0}, {1, y < 0}}]Show[Histogram[dets, {0.1}, PDF], Plot[detpdf[y], {y, -5, 4}, Exclusions -> None]]Estimate the spectral density of matrix from GaussianUnitaryMatrixDistribution:
spectrum𝒟[n_] := MatrixPropertyDistribution[Eigenvalues[x], xGaussianUnitaryMatrixDistribution[n]];The closed form is known to be the following:
spectralPDF[n_Integer, y_] := Sqrt[(2/π n)]Exp[-2n y ^ 2]Sum[(1/2^jj!)HermiteH[j, Sqrt[2n] y]^2, {j, 0, n - 1}]For smaller matrices, there is a characteristic oscillatory pattern:
n = 3;
eigvs = Join@@RandomVariate[spectrum𝒟[n], 10 ^ 5];Compare the histogram of the sample to the known PDF:
Show[Histogram[eigvs / (2Sqrt[n]), {0.05}, PDF], Plot[spectralPDF[n, x]//Evaluate, {x, -1.5, 1.5}]]The number of density maxima is equal to the matrix size:
n = 4;
eigvs = Join@@RandomVariate[spectrum𝒟[n], 10 ^ 5];Show[Histogram[eigvs / (2Sqrt[n]), {0.05}, PDF], Plot[spectralPDF[n, x]//Evaluate, {x, -1.5, 1.5}]]In the limit of large matrices, the density converges to WignerSemicircleDistribution:
n = 250;
eigvs = Join@@RandomVariate[spectrum𝒟[n], 10 ^ 2];Show[Histogram[eigvs / (2Sqrt[n]), {0.05}, PDF], Plot[PDF[WignerSemicircleDistribution[1], x], {x, -1.5, 1.5}, Exclusions -> None]]The zeros of the Riemann zeta function have been conjectured to be related to the eigenvalues of Hermitian operators and matrices. Compare the normalized spacing of the zeros to the normalized spacing of the bulk eigenvalues of samples from GaussianUnitaryMatrixDistribution:
n = 10 ^ 2;
eigs = RandomVariate[MatrixPropertyDistribution[Take[Sort[Eigenvalues[x]], {1, 3}Quotient[n, 4]], xGaussianUnitaryMatrixDistribution[n]], 10 ^ 3];Compare the histogram of normalized spacings to the known PDF:
spacings = Flatten[(Differences[#] Sqrt[4 n - Most[#] ^ 2]& /@ eigs) / (2 Pi)];WignerSurmise[x_, 2] := 2(4 x / Pi) ^ 2 Exp[(-4 / Pi) x ^ 2]Show[Histogram[spacings, 50, PDF], Plot[WignerSurmise[x, 2], {x, 0, 2.5}]]Compare this PDF to the normalized spacing for the zeros of the zeta function in the critical line for a series of the zeros starting at the 
zero (Odzlyko):
zeros = TemporalData[EventSeries, {{{2.6765339564884753*^11, 2.6765339564936237*^11,
2.6765339564968164*^11, 2.67653395649862*^11, 2.6765339565015765*^11, 2.6765339565043427*^11,
2.676533956505809*^11, 2.6765339565083447*^11, 2.6765339565105847*^ ... 676533982146852*^11, 2.6765339821491986*^11, 2.6765339821520862*^11, 2.6765339821532565*^11,
2.676533982157319*^11, 2.6765339821604626*^11}}, {{0, 9999, 1}}, 1, {"Discrete", 1},
{"Discrete", 1}, 1, {ResamplingMethod -> None}}, False, 10.3];
zlist = zeros["Values"];Check that they indeed are zeros:
{#, RiemannSiegelZ[#]}& /@ RandomChoice[zlist, 3]Compare the histogram of normalized spacings to the known PDF for the random matrices:
Show[Histogram[(Differences[zlist]/2 Pi) Log[(Most[zlist]/2 Pi)], 50, PDF], Plot[WignerSurmise[x, 2], {x, 0, 2.5}]]Define distribution for scaled condition number of a WishartMatrixDistribution:
n = 200;
wmd = WishartMatrixDistribution[n, IdentityMatrix[n, SparseArray]];
cn𝒟 = MatrixPropertyDistribution[(1/n)Sqrt[(Max[#]/Min[#])]&[Eigenvalues[𝓂]], 𝓂wmd];Sample the scaled condition number of a large matrix and check that it agrees with asymptotic closed-form distribution:
asymp𝒟 = ProbabilityDistribution[(2κ + 4/κ^3)Exp[-(2/κ) - (2/κ^2)], {κ, 0, Infinity}];data = RandomVariate[cn𝒟, 2000];Show[
Histogram[data, {0.8}, PDF],
Plot[PDF[asymp𝒟, κ], {κ, 0.1, 30}, PlotRange -> All]]The asymptotic scaled condition number distribution has infinite mean:
generalizedMean = Expectation[κ^α, κasymp𝒟, Assumptions -> 0 < α < 1]meanScaledCN = Limit[generalizedMean, α -> 1, Direction -> 1]Simulate whether LinearSolve determines the random matrix to be ill‐conditioned:
lucQ[m_] := Quiet[Check[LinearSolve[m];0, 1], LinearSolve::luc]condIndicators =
RandomVariate[MatrixPropertyDistribution[lucQ[𝓂], 𝓂wmd], 5000];Infer the probability that the random Wishart matrix is badly conditioned:
FindDistributionParameters[condIndicators, BernoulliDistribution[p]]Use asymptotic distribution to infer the critical ratio of the largest and the smallest eigenvalues:
(n InverseSurvivalFunction[asymp𝒟, p /. %]) ^ 2Text
Wolfram Research (2015), MatrixPropertyDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/MatrixPropertyDistribution.html.
CMS
Wolfram Language. 2015. "MatrixPropertyDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MatrixPropertyDistribution.html.
APA
Wolfram Language. (2015). MatrixPropertyDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MatrixPropertyDistribution.html