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

$S_{p\times p}=\sum _{i=1}^n{G}_i{G}_i^H$

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 $𝔼[G{G}^H]=M$ then

$S\sim n𝒞𝒲(M,n,p)$

where $𝒞𝒲(M,n,p)$ is the complex central Wishart distribution with n degrees of freedom and mean value, or scale matrix, M.

$f_S(𝐒)=\frac{{\left|𝐒\right|}^{n-p}{e}^{-\mathrm{tr}({𝐌}^{-1}𝐒)}}{{\left|𝐌\right|}^n\cdot 𝒞{\tilde{\mathrm{\Gamma }}}_p(n)},n\ge p,\left|𝐌\right|>0$

where

$𝒞{\tilde{\mathrm{\Gamma }}}_p^{}(n)={\pi }^{p(p-1)/2}\prod _{j=1}^p\mathrm{\Gamma }(n-j+1)$

is the complex multivariate Gamma function.

Using the trace rotation rule $\mathrm{tr}(ABC)=\mathrm{tr}(CAB)$ we also get

$f_S(𝐒)=\frac{{\left|𝐒\right|}^{n-p}}{{\left|𝐌\right|}^n\cdot 𝒞{\tilde{\mathrm{\Gamma }}}_p(n)}\exp (-\sum _{i=1}^p{G}_i^H{𝐌}^{-1}{G}_i)$

which is quite close to the complex multivariate pdf of G itself. The elements of G conventionally have circular symmetry such that $𝔼[G{G}^T]=0$.

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

$f_Y(𝐘)=\frac{{\left|𝐘\right|}^{-(n+p)}{e}^{-\mathrm{tr}(𝐌{𝐘}^{\mathbf{-}\mathrm{𝟏}})}}{{\left|𝐌\right|}^{-n}\cdot 𝒞{\tilde{\mathrm{\Gamma }}}_p(n)},n\ge p,\det \left(𝐘\right)>0$

where $𝐌={\mathrm{\Gamma }}^{\mathbf{-}\mathrm{𝟏}}$.

If derived via a matrix inversion mapping, the result depends on the complex Jacobian determinant

$𝒞J_Y(Y^{-1})={\left|Y\right|}^{-2p-2}$

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

$f({\lambda }_1\dots {\lambda }_p)={\tilde{K}}_{\nu ,p}\exp (-\frac{1}{2}\sum _{i=1}^p{\lambda }_i)\prod _{i=1}^p{\lambda }_i^{\nu -p}\prod _{i<j}({\lambda }_i-{\lambda }_j{)}^{2}d{\lambda }_1\dots d{\lambda }_p,{\lambda }_i\in ℝ\ge 0$

where

${\tilde{K}}_{\nu ,p}^{-1}=2^{p\nu }\prod _{i=1}^p\mathrm{\Gamma }(\nu -i+1)\mathrm{\Gamma }(p-i+1)$

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 $Z=𝐄(X^2+Y^2)=2$. 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 $S_{p\times p}\sim 𝒞𝒲(2𝐈,\frac{p}{\kappa })$ then in the limit $p\to \mathrm{\infty }$ 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 $S_{p\times p}\sim 𝒞𝒲(𝐈,\frac{p}{\kappa })$, 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

$p_y(y)=\frac{1}{2\pi }{(4-y^2)}^{\frac{1}{2}},-2\le y\le 2$

In place of the definition of the Wishart sample matrix above, $S_{p\times p}=\sum _{j=1}^{\nu }{G}_j{G}_j^H$, we can define a Gaussian ensemble

${𝐆}_{i,j}=[G_1\dots G_{\nu }]\in {ℂ}^{p\times \nu }$

such that S is the matrix product $S=𝐆{𝐆}^{𝐇}$. The real non-negative eigenvalues of S are then the modulus-squared singular values of the ensemble $𝐆$ 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 $𝐆$ are invariant under transposition so, redefining $\tilde{S}={𝐆}^{𝐇}𝐆$, 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 $𝐆$ 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

$𝐆=Q\left[\begin{array}{c}𝐑\\ 0\end{array}\right]$

such that ${𝐑}_{q\times q},q\le \nu$ is upper triangular with full rank and ${\widetilde{\stackrel{~}{S}}}_{q\times q}={𝐑}^{𝐇}𝐑$ 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.