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
 * $$ S_{p \times p} = \sum_{i=1}^n G_iG_i^H $$

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 $$ \mathbb{E}[GG^H] = M $$ then
 * $$ S \sim n\mathcal{CW}(M,n,p) $$

where $$ \mathcal{CW}(M,n,p) $$ is the complex central Wishart distribution with n degrees of freedom and mean value, or scale matrix, M.
 * $$ f_S(\mathbf{S}) = \frac{

\left |\mathbf{S} \right|^{n-p} e^{-\operatorname{tr}(\mathbf M^{-1}\mathbf{S}) } } { \left|\mathbf{M}\right|^n\cdot \mathcal{C} \widetilde{\Gamma}_p(n) }, \;\;\; n\ge p, \;\;\;  \left|\mathbf{M}\right| > 0$$ where
 * $$ \mathcal{C} \widetilde{\Gamma}_p^{} (n) = \pi^{p(p-1)/2} \prod_{j=1}^p \Gamma (n-j+1) $$

is the complex multivariate Gamma function.

Using the trace rotation rule $$ \operatorname{tr}(ABC) = \operatorname{tr}(CAB) $$ we also get
 * $$ f_S(\mathbf{S}) = \frac{

\left |\mathbf{S} \right|^{n-p} } { \left|\mathbf{M}\right|^n\cdot \mathcal{C} \widetilde{\Gamma}_p(n) } \exp \left( -\sum_{i=1}^p G_i^H\mathbf M^{-1} 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{E}[GG^T] = 0 $$.

Inverse Complex Wishart The distribution of the inverse complex Wishart distribution of $$ \mathbf{Y} = \mathbf{S^{-1}} $$ according to Goodman, Shaman is
 * $$ f_Y(\mathbf{Y}) = \frac{

\left |\mathbf{Y} \right|^{-(n+p)} e^{-\operatorname{tr}(\mathbf M\mathbf{Y^{-1}}) } } { \left|\mathbf{M}\right|^{-n}\cdot\mathcal{C}\widetilde{\Gamma}_p(n) }, \;\;\; n\ge p, \;\;\;  \det \left(\mathbf{Y}\right) > 0$$ where $$ \mathbf{M} = \mathbf{\Gamma^{-1}}$$.

If derived via a matrix inversion mapping, the result depends on the complex Jacobian determinant
 * $$ \mathcal{C}J_Y(Y^{-1}) = \left | Y \right |^{-2p-2} $$

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 {K}_{\nu,p} \exp \left ( - \frac{1}{2} \sum_{i=1}^p \lambda_i \right )

\prod_{i=1}^p \lambda_i^{\nu - p} \prod_{i<j} (\lambda_i - \lambda_j)^2 d\lambda_1 \dots d\lambda_p, \;\;\; \lambda_i \in \mathbb{R} \ge 0$$ where
 * $$ \tilde {K}_{\nu,p}^{-1} = 2^{p\nu} \prod_{i=1}^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 $$ Z = \mathbf{E} \left(X^2 + Y^2 \right ) = 2$$. 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 $$ S_{p \times p} \sim \mathcal{CW}\left( 2\mathbf{I}, \frac{p}{\kappa} \right) $$ then in the limit $$ p \rightarrow \infty $$ the distribution of eigenvalues converges in probability to the Marchenko–Pastur distribution function
 * $$ p_\lambda(\lambda) = \frac

{\sqrt { [\lambda/2 - ( \sqrt {\kappa} -1 )^2 ][\sqrt {\kappa} +1 )^2 - \lambda /2 ] }} { 4\pi \kappa (\lambda /2)}, \;\;\; 2( \sqrt {\kappa} -1)^2 \le \lambda \le  2(\sqrt {\kappa} +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_{p \times p} \sim \mathcal{CW} \left( \mathbf{I}, \frac{p}{\kappa} \right) $$, the pdf reduces to the real Wishart one:
 * $$ p_\lambda(\lambda) = \frac

{\sqrt {[\lambda - ( \sqrt {\kappa} -1 )^2 ][\sqrt {\kappa} +1 )^2 - \lambda ] }} { 2\pi \kappa \lambda}, \;\;\;   (\sqrt {\kappa} -1)^2 \le \lambda \le  (\sqrt {\kappa} +1 )^2, \;\;\; 0 \le \kappa \le 1 $$

A special case is $$ \kappa = 1 $$


 * $$ p_\lambda(\lambda) = \frac {1}{4\pi} \left (\frac {8-\lambda}{\lambda} \right )^{\frac{1}{2}}, \; 0 \le \lambda \le 8 $$

or, if a Var(Z) = 1 convention is used then
 * $$ p_\lambda(\lambda) = \frac {1}{2\pi} \left (\frac {4-\lambda}{\lambda} \right )^{\frac{1}{2}}, \; 0 \le \lambda \le 4 $$.

The Wigner semicircle distribution arises by making the change of variable $$ y = \pm\sqrt{\lambda} $$ in the latter and selecting the sign of y randomly yielding pdf
 * $$ p_y(y) = \frac {1}{2\pi} \left ( 4-y^2 \right )^{\frac{1}{2}}, \; -2 \le y \le 2 $$

In place of the definition of the Wishart sample matrix above, $$ S_{p \times p} = \sum_{j=1}^\nu G_jG_j^H $$, we can define a Gaussian ensemble
 * $$ \mathbf{G}_{i,j} = [G_1 \dots G_\nu ] \in \mathbb{C}^{\,p \times \nu } $$

such that S is the matrix product $$ S = \mathbf{G}\mathbf{G^H} $$. The real non-negative eigenvalues of S are then the modulus-squared singular values of the ensemble $$ \mathbf{G}$$ 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 $$ \mathbf{G} $$ are invariant under transposition so, redefining $$ \tilde{S} = \mathbf{G^H}\mathbf{G}  $$, then $$ \tilde{S}_{\nu \times \nu} $$ has a complex Wishart distribution, has full rank almost certainly, and eigenvalue distributions can be obtained from $$ \tilde{S} $$ in lieu, using all the previous equations.

In cases where the columns of $$ \mathbf{G} $$ are not linearly independent and $$ \tilde{S}_{\nu \times \nu} $$ remains singular, a QR decomposition can be used to reduce G to a product like

\mathbf{G} =  Q \begin{bmatrix} \mathbf{R} \\ 0 \end{bmatrix} $$ such that $$ \mathbf{R}_{q \times q}, \;\; q \le \nu  $$ is upper triangular with full rank and $$ \tilde\tilde{S}_{q \times q} = \mathbf{R^H}\mathbf{R}  $$ 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.