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Complex Wishart
Notation	A ~CWp(${\displaystyle \mathrm{\Gamma }}$,n)
Parameters	n >p − 1degrees of freedom (real) ${\displaystyle \mathrm{\Gamma }}$ > 0 (p ×pHermitianpos. def)
Support	A (p ×p)Hermitianpositive definite matrix
PDF	${\displaystyle \frac{det{\left(\mathbf{A}\right)}^{(n-p)}{e}^{-\mathrm{tr}({\mathbf{\Gamma }}^{-1}\mathbf{A})}}{det{\left(\mathbf{\Gamma }\right)}^n\cdot \mathcal{C}{\tilde{\mathrm{\Gamma }}}_p(n)}}$ ${\displaystyle \mathcal{C}{\tilde{\mathbf{\Gamma }}}_p}$ is the${\displaystyle p}$-variatecomplex multivariate gamma function tr is thetrace function
Mean	${\displaystyle \mathrm{E}[A]=n\mathrm{\Gamma }}$
Mode	${\displaystyle (n-p)\mathbf{\Gamma }}$ forn ≥p + 1
CF	${\displaystyle det{\left(I_p-i\mathbf{\Gamma }\mathbf{\Theta }\right)}^{-n}}$

Instatistics, thecomplex Wishart distribution is acomplex version of theWishart distribution. It is the distribution of${\displaystyle n}$ times the sample Hermitian covariance matrix of${\displaystyle n}$ zero-meanindependentGaussian random variables. It hassupport for${\displaystyle p\times p}$Hermitianpositive definite matrices.^[1]

The complex Wishart distribution is also encountered in wireless communications, while analyzing the performance ofRayleigh fadingMIMO wireless channels.^[2]

The complex Wishart distribution is the density of a complex-valued sample covariance matrix. Let

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

where each${\displaystyle G_i}$ is an independent columnp-vector of random complex Gaussian zero-mean samples and${\displaystyle (.{)}^H}$ is an Hermitian (complex conjugate) transpose. If the covariance ofG is${\displaystyle \mathbb{E}[G{G}^H]=M}$ then

${\displaystyle S\sim n\mathcal{C}\mathcal{W}(M,n,p)}$

where${\displaystyle \mathcal{C}\mathcal{W}(M,n,p)}$ is the complex central Wishart distribution withn degrees of freedom and mean value, or scale matrix,M.

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

where

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

is the complex multivariate Gamma function.^[3]

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

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

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

Inverse Complex WishartThe distribution of the inverse complex Wishart distribution of${\displaystyle \mathbf{Y}={\mathbf{S}}^{\mathbf{-}\mathbf{1}}}$ according to Goodman,^[3] Shaman^[4] is

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

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

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

${\displaystyle \mathcal{C}{J}_Y(Y^{-1})={\left|Y\right|}^{-2p}}$

Goodman and others^[5] discuss such complex Jacobians.

The probability distribution of the eigenvalues of the complex Hermitian Wishart distribution are given by, for example, James^[6] and Edelman.^[7] For a${\displaystyle p\times p}$ matrix with${\displaystyle \nu \ge p}$ degrees of freedom we have

where

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

Note however that Edelman uses the "mathematical" definition of a complex normal variable${\displaystyle Z=X+iY}$ where iidX andY each have unit variance and the variance of${\displaystyle Z=\mathbf{E}\left(X^2+Y^2\right)=2}$. For the definition more common in engineering circles, withX andY each having 0.5 variance, the eigenvalues are reduced by a factor of 2.

This spectral density can be integrated to give the probability that all eigenvalues of a Wishart random matrix lie within an interval.^[8]

The spectral density can be also integrated to give the marginal distribution of eigenvalues.^[9][10]

There are also approximations for marginal eigenvalue distributions. From Edelman we have that ifS is a sample from the complex Wishart distribution with${\displaystyle p=\kappa \nu ,0\le \kappa \le 1}$ such that${\displaystyle S_{p\times p}\sim \mathcal{C}\mathcal{W}\left(2\mathbf{I},\frac{p}{\kappa }\right)}$then in the limit${\displaystyle p\to \mathrm{\infty }}$ the distribution of eigenvalues converges in probability to theMarchenko–Pastur distribution function

${\displaystyle p_{\lambda }(\lambda )=\frac{\sqrt{[\lambda /2-(\sqrt{\kappa }-1{)}^2][\sqrt{\kappa }+1{)}^2-\lambda /2]}}{4\pi \kappa (\lambda /2)},2(\sqrt{\kappa }-1{)}^2\le \lambda \le 2(\sqrt{\kappa }+1{)}^2,0\le \kappa \le 1}$

This distribution becomes identical to the real Wishart case, by replacing${\displaystyle \lambda }$ by${\displaystyle 2\lambda }$, on account of the doubled sample variance, so in the case${\displaystyle S_{p\times p}\sim \mathcal{C}\mathcal{W}\left(\mathbf{I},\frac{p}{\kappa }\right)}$, the pdf reduces to the real Wishart one:

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

A special case is${\displaystyle \kappa =1}$

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

or, if a Var(Z) = 1 convention is used then

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

TheWigner semicircle distribution arises by making the change of variable${\displaystyle y=\pm \sqrt{\lambda }}$ in the latter and selecting the sign ofy randomly yielding pdf

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

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

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

such thatS is the matrix product${\displaystyle S=\mathbf{G}{\mathbf{G}}^{\mathbf{H}}}$. The real non-negative eigenvalues ofS are then themodulus-squared singular values of the ensemble${\displaystyle \mathbf{G}}$ and the moduli of the latter have a quarter-circle distribution.

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

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

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

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

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

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