Complex Wishart Distribution

In statistics, the complex Wishart distribution is a complex version of the Wishart distribution. It is the distribution of $n$ times the sample Hermitian covariance matrix of $n$ zero-mean independent Gaussian random variables. It has support for $p\times p$ Hermitian positive definite matrices.

The complex Wishart distribution is the density of a complex-valued sample covariance matrix. Let
:$S_ = \sum_^n G_iG_i^H$
where each $G_i$ is an independent column ''p''-vector of random complex Gaussian zero-mean samples and $(.)^H$ is an Hermitian (complex conjugate) transpose. If the covariance of ''G'' is \mathbb G^H= M then
:$S \sim n\mathcal(M,n,p)$
where $\mathcal(M,n,p)$ is the complex central Wishart distribution with ''n'' degrees of freedom and mean value, or scale matrix, ''M''.
:$f_S(\mathbf) = \frac , \;\;\; n\ge p, \;\;\; \left, \mathbf\ > 0$
where
:$\mathcal \widetilde_p^ (n) = \pi^ \prod_^p \Gamma (n-j+1)$
is the complex multivariate Gamma function.

Using the trace rotation rule $\operatorname(ABC) = \operatorname(CAB)$ we also get
:$f_S(\mathbf) = \frac \exp \left( -\sum_^p G_i^H\mathbf M^ G_i \right )$
which is quite close to the complex multivariate pdf of ''G'' itself. The elements of ''G'' conventionally have circular symmetry such that \mathbb G^T= 0 .

Inverse Complex Wishart
The distribution of the inverse complex Wishart distribution of $\mathbf = \mathbf$ according to Goodman, Shaman is
:$f_Y(\mathbf) = \frac , \;\;\; n\ge p, \;\;\; \det \left(\mathbf\right) > 0$
where $\mathbf = \mathbf$.

If derived via a matrix inversion mapping, the result depends on the complex Jacobian determinant
:$\mathcalJ_Y(Y^) = \left , Y \right , ^$

Goodman and others discuss such complex Jacobians.

Eigenvalues

The probability distribution of the eigenvalues of the complex Hermitian Wishart distribution are given by, for example, James and Edelman. For a $p \times p$ matrix with $\nu \ge p$ degrees of freedom we have
:$f(\lambda_1\dots\lambda_p)=\tilde _ \exp \left ( - \frac \sum_^p \lambda_i \right ) \prod_^p \lambda_i^ \prod_ (\lambda_i - \lambda_j)^2 d\lambda_1 \dots d\lambda_p, \;\;\; \lambda_i \in \mathbb \ge 0$
where
:$\tilde _^ = 2^ \prod_^p \Gamma (\nu - i+1) \Gamma (p-i+1)$
Note however that Edelman uses the "mathematical" definition of a complex normal variable $Z = X + iY$ where iid ''X'' and ''Y'' each have unit variance and the variance of $Z = \mathbf \left(X^2 + Y^2 \right ) = 2$. 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 $p = \kappa \nu, \;\; 0 \le \kappa \le 1$ such that $S_ \sim \mathcal\left( 2\mathbf, \frac \right)$
then in the limit $p \rightarrow \infty$ the distribution of eigenvalues converges in probability to the Marchenko–Pastur distribution function
:$p_\lambda(\lambda) = \frac , \;\;\; 2( \sqrt -1)^2 \le \lambda \le 2(\sqrt +1 )^2, \;\;\; 0 \le \kappa \le 1$
This distribution becomes identical to the real Wishart case, by replacing $\lambda$ by $2\lambda$, on account of the doubled sample variance, so in the case $S_ \sim \mathcal \left( \mathbf, \frac \right)$, the pdf reduces to the real Wishart one:
:$p_\lambda(\lambda) = \frac , \;\;\; (\sqrt -1)^2 \le \lambda \le (\sqrt +1 )^2, \;\;\; 0 \le \kappa \le 1$

A special case is $\kappa = 1$

:$p_\lambda(\lambda) = \frac \left (\frac \right )^, \; 0 \le \lambda \le 8$
or, if a Var(''Z'') = 1 convention is used then
:$p_\lambda(\lambda) = \frac \left (\frac \right )^, \; 0 \le \lambda \le 4$.
The Wigner semicircle distribution arises by making the change of variable $y = \pm\sqrt$ in the latter and selecting the sign of ''y'' randomly yielding pdf
:$p_y(y) = \frac \left ( 4-y^2 \right )^, \; -2 \le y \le 2$

In place of the definition of the Wishart sample matrix above, $S_ = \sum_^\nu G_jG_j^H$, we can define a Gaussian ensemble
: \mathbf_ = _1 \dots G_\nu \in \mathbb^
such that ''S'' is the matrix product $S = \mathbf\mathbf$. The real non-negative eigenvalues of ''S'' are then the modulus-squared singular values of the ensemble $\mathbf$ and the moduli of the latter have a quarter-circle distribution.

In the case $\kappa > 1$ such that $\nu < p$ then $S$ is rank deficient with at least $p - \nu$ null eigenvalues. However the singular values of $\mathbf$ are invariant under transposition so, redefining $\tilde = \mathbf\mathbf$, then $\tilde_$ has a complex Wishart distribution, has full rank almost certainly, and eigenvalue distributions can be obtained from $\tilde$ in lieu, using all the previous equations.

In cases where the columns of $\mathbf$ are not linearly independent and $\tilde_$ remains singular, a QR decomposition can be used to reduce ''G'' to a product like
:$\mathbf = Q \begin \mathbf \\ 0 \end$
such that $\mathbf_, \;\; q \le \nu$ is upper triangular with full rank and $\tilde\tilde_ = \mathbf\mathbf$ has further reduced dimensionality.

The eigenvalues are of practical significance in radio communications theory since they define the Shannon channel capacity of a $\nu \times p$ MIMO wireless channel which, to first approximation, is modeled as a zero-mean complex Gaussian ensemble.

References

{{DEFAULTSORT:Complex Wishart Distribution
Continuous distributions
Multivariate continuous distributions
Covariance and correlation
Random matrices
Conjugate prior distributions
Exponential family distributions
Complex distributions