Complex Wishart distribution

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 ${\displaystyle \nu \geq p}$ degrees of freedom we have

${\displaystyle f(\lambda _{1}\dots \lambda _{p})={\tilde {K}}_{\nu ,p}\exp \left(-{\frac {1}{2}}\sum _{i=1}^{p}\lambda _{i}\right)\prod _{i=1}^{p}\lambda _{i}^{\nu -p}\prod _{i<j}(\lambda _{i}-\lambda _{j})^{2}d\lambda _{1}\dots d\lambda _{p},\;\;\;\lambda _{i}\in \mathbb {R} \geq 0}$

where

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

Note however that Edelman uses the "mathematical" definition of a complex normal variable ${\displaystyle Z=X+iY}$ where iid X and Y 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, with X and Y 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 if S is a sample from the complex Wishart distribution with ${\displaystyle p=\kappa \nu ,\;\;0\leq \kappa \leq 1}$ such that ${\displaystyle S_{p\times p}\sim {\mathcal {CW}}\left(2\mathbf {I} ,{\frac {p}{\kappa }}\right)}$ then in the limit ${\displaystyle p\rightarrow \infty }$ the distribution of eigenvalues converges in probability to the Marchenko–Pastur distribution function

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

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 ${\displaystyle S_{p\times p}\sim {\mathcal {CW}}\left(\mathbf {I} ,{\frac {p}{\kappa }}\right)}$, the pdf reduces to the real Wishart one:

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

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

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

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

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

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

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

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

${\displaystyle \mathbf {G} _{i,j}=[G_{1}\dots G_{\nu }]\in \mathbb {C} ^{\,p\times \nu }}$

such that S is the matrix product ${\displaystyle S=\mathbf {G} \mathbf {G^{H}} }$. The real non-negative eigenvalues of S are then the modulus-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 ${\displaystyle \nu <p}$ 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 ${\displaystyle {\tilde {S}}=\mathbf {G^{H}} \mathbf {G} }$, 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, a QR decomposition can be used to reduce G to a product like

${\displaystyle \mathbf {G} =Q{\begin{bmatrix}\mathbf {R} \\0\end{bmatrix}}}$

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

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