Notation A ~ CWp($\Gamma$ , n) n > p − 1 degrees of freedom (real)$\Gamma$ > 0 (p × p Hermitian pos. def) A (p × p) Hermitian positive definite matrix ${\frac {\det \left(\mathbf {A} \right)^{(n-p)}e^{-\operatorname {tr} (\mathbf {\Gamma } ^{-1}\mathbf {A} ))){\det \left(\mathbf {\Gamma } \right)^{n}\cdot {\mathcal {C)){\widetilde {\Gamma ))_{p}(n)))$ ${\mathcal {C)){\widetilde {\mathbf {\Gamma } ))_{p)$ is the $p$ -variate complex multivariate gamma function tr is the trace function $\operatorname {E} [A]=n\Gamma$ $(n-p)\mathbf {\Gamma }$ for n ≥ p + 1 $\det \left(I_{p}-i\mathbf {\Gamma } \mathbf {\Theta } \right)^{-n)$ 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 $\mathbb {E} [GG^{H}]=M$ 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.

$f_{S}(\mathbf {S} )={\frac {\left|\mathbf {S} \right|^{n-p}e^{-\operatorname {tr} (\mathbf {M} ^{-1}\mathbf {S} ))){\left|\mathbf {M} \right|^{n}\cdot {\mathcal {C)){\widetilde {\Gamma ))_{p}(n))),\;\;\;n\geq p,\;\;\;\left|\mathbf {M} \right|>0$ 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

$f_{Y}(\mathbf {Y} )={\frac {\left|\mathbf {Y} \right|^{-(n+p)}e^{-\operatorname {tr} (\mathbf {M} \mathbf {Y^{-1)) ))){\left|\mathbf {M} \right|^{-n}\cdot {\mathcal {C)){\widetilde {\Gamma ))_{p}(n))),\;\;\;n\geq p,\;\;\;\det \left(\mathbf {Y} \right)>0$ 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 $\nu \geq p$ 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 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 $p=\kappa \nu ,\;\;0\leq \kappa \leq 1$ 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

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

$p_{\lambda }(\lambda )={\frac {\sqrt {[\lambda -({\sqrt {\kappa ))-1)^{2}][{\sqrt {\kappa ))+1)^{2}-\lambda ])){2\pi \kappa \lambda )),\;\;\;({\sqrt {\kappa ))-1)^{2}\leq \lambda \leq ({\sqrt {\kappa ))+1)^{2},\;\;\;0\leq \kappa \leq 1$ A special case is $\kappa =1$ $p_{\lambda }(\lambda )={\frac {1}{4\pi ))\left({\frac {8-\lambda }{\lambda ))\right)^{\frac {1}{2)),\;0\leq \lambda \leq 8$ or, if a Var(Z) = 1 convention is used then

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

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

$p_{y}(y)={\frac {1}{2\pi ))\left(4-y^{2}\right)^{\frac {1}{2)),\;-2\leq y\leq 2$ 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

$\mathbf {G} _{i,j}=[G_{1}\dots G_{\nu }]\in \mathbb {C} ^{\,p\times \nu )$ 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 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 $\mathbf {R} _{q\times q},\;\;q\leq \nu$ 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.

1. ^ N. R. Goodman (1963). "The distribution of the determinant of a complex Wishart distributed matrix". The Annals of Mathematical Statistics. 34 (1): 178–180. doi:10.1214/aoms/1177704251.
2. ^ a b Goodman, N R (1963). "Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction)". Ann. Math. Statist. 34: 152–177. doi:10.1214/aoms/1177704250.
3. ^ Shaman, Paul (1980). "The Inverted Complex Wishart Distribution and Its Application to Spectral Estimation". Journal of Multivariate Analysis. 10: 51–59. doi:10.1016/0047-259X(80)90081-0.
4. ^ Cross, D J (May 2008). "On the Relation between Real and Complex Jacobian Determinants" (PDF). drexel.edu.
5. ^ James, A. T. (1964). "Distributions of Matrix Variates and Latent Roots Derived from Normal Samples". Ann. Math. Statist. 35 (2): 475–501. doi:10.1214/aoms/1177703550.
6. ^ Edelman, Alan (October 1988). "Eigenvalues and Condition Numbers of Random Matrices" (PDF). SIAM J. Matrix Anal. Appl. 9 (4): 543–560. doi:10.1137/0609045. hdl:1721.1/14322.