In statistics, the complex Wishart distribution is a complex version of the Wishart distribution. It is the distribution of ${\displaystyle n}$ times the sample Hermitian covariance matrix of ${\displaystyle n}$ zero-mean independent Gaussian random variables. It has support for ${\displaystyle p\times p}$ Hermitian positive definite matrices.

The complex Wishart distribution is also encountered in wireless communications, while analyzing the performance of Rayleigh fading MIMO wireless channels.

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

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

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

where

is the complex multivariate Gamma function.

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