In statistics, the complex Wishart distribution is a complex version of the Wishart distribution.

times the sample Hermitian covariance matrix of

zero-mean independent Gaussian random variables.

Hermitian positive definite matrices.

[1] The complex Wishart distribution is the density of a complex-valued sample covariance matrix.

is an independent column p-vector of random complex Gaussian zero-mean samples and

is an Hermitian (complex conjugate) transpose.

is the complex central Wishart distribution with n degrees of freedom and mean value, or scale matrix, M. where is the complex multivariate Gamma function.

we also get which is quite close to the complex multivariate pdf of G itself.

The elements of G conventionally have circular symmetry such that

Inverse Complex Wishart The distribution of the inverse complex Wishart distribution of

If derived via a matrix inversion mapping, the result depends on the complex Jacobian determinant Goodman and others[4] discuss such complex Jacobians.

The probability distribution of the eigenvalues of the complex Hermitian Wishart distribution are given by, for example, James[5] and Edelman.

degrees of freedom we have where Note however that Edelman uses the "mathematical" definition of a complex normal variable

For the definition more common in engineering circles, with X and Y each having 0.5 variance, the eigenvalues are reduced by a factor of 2.

While this expression gives little insight, there are approximations for marginal eigenvalue distributions.

From Edelman we have that if S is a sample from the complex Wishart distribution with

the distribution of eigenvalues converges in probability to the Marchenko–Pastur distribution function This distribution becomes identical to the real Wishart case, by replacing

, on account of the doubled sample variance, so in the case

, the pdf reduces to the real Wishart one: A special case is

or, if a Var(Z) = 1 convention is used then The Wigner semicircle distribution arises by making the change of variable

in the latter and selecting the sign of y randomly yielding pdf In place of the definition of the Wishart sample matrix above,

, we can define a Gaussian ensemble such that S is the matrix product

The real non-negative eigenvalues of S are then the modulus-squared singular values of the ensemble

and the moduli of the latter have a quarter-circle distribution.

has a complex Wishart distribution, has full rank almost certainly, and eigenvalue distributions can be obtained from

remains singular, a QR decomposition can be used to reduce G to a product like such that

is upper triangular with full rank and

The eigenvalues are of practical significance in radio communications theory since they define the Shannon channel capacity of a

MIMO wireless channel which, to first approximation, is modeled as a zero-mean complex Gaussian ensemble.