Definitions for complex wishart distribution
com·plex wishart dis·tri·bu·tion

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1. Complex Wishart distribution

   In statistics, the complex Wishart distribution is a complex version of the Wishart distribution. It is the distribution of n {\displaystyle n} times the sample Hermitian covariance matrix of n {\displaystyle n} zero-mean independent Gaussian random variables. It has support for p × p {\displaystyle p\times p} Hermitian positive definite matrices.The complex Wishart distribution is the density of a complex-valued sample covariance matrix. Let S p × p = ∑ i = 1 n G i G i H {\displaystyle S_{p\times p}=\sum _{i=1}^{n}G_{i}G_{i}^{H}} where each G i {\displaystyle G_{i}} is an independent column p-vector of random complex Gaussian zero-mean samples and ( . ) H {\displaystyle (.)^{H}} is an Hermitian (complex conjugate) transpose. If the covariance of G is E [ G G H ] = M {\displaystyle \mathbb {E} [GG^{H}]=M} then S ∼ n C W ( M , n , p ) {\displaystyle S\sim n{\mathcal {CW}}(M,n,p)} where C W ( M , n , p ) {\displaystyle {\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 ( S ) = | S | n − p e − tr ⁡ ( M − 1 S ) | M | n ⋅ C Γ ~ p ( n ) , n ≥ p , | M | > 0 {\displaystyle 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 C Γ ~ p ( n ) = π p ( p − 1 ) / 2 ∏ j = 1 p Γ ( n − j + 1 ) {\displaystyle {\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 tr ⁡ ( A B C ) = tr ⁡ ( C A B ) {\displaystyle \operatorname {tr} (ABC)=\operatorname {tr} (CAB)} we also get f S ( S ) = | S | n − p | M | n ⋅ C Γ ~ p ( n ) exp ⁡ ( − ∑ i = 1 p G i H M − 1 G i ) {\displaystyle 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 E [ G G T ] = 0 {\displaystyle \mathbb {E} [GG^{T}]=0} Inverse Complex Wishart The distribution of the inverse complex Wishart distribution of Y = S − 1 {\displaystyle \mathbf {Y} =\mathbf {S^{-1}} } according to Goodman, Shaman is f Y ( Y ) = | Y | − ( n + p ) e − tr ⁡ ( M Y − 1 ) | M | − n ⋅ C Γ ~ p ( n ) , n ≥ p , det ( Y ) > 0 {\displaystyle 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 M = Γ − 1 {\displaystyle \mathbf {M} =\mathbf {\Gamma ^{-1}} } . If derived via a matrix inversion mapping, the result depends on the complex Jacobian determinant C J Y ( Y − 1 ) = | Y | − 2 p − 2 {\displaystyle {\mathcal {C}}J_{Y}(Y^{-1})=\left|Y\right|^{-2p-2}} Goodman and others discuss such complex Jacobians.

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Numerology

1. Chaldean Numerology

   The numerical value of complex wishart distribution in Chaldean Numerology is: 7

2. Pythagorean Numerology

   The numerical value of complex wishart distribution in Pythagorean Numerology is: 4

References

1. ^ Wikipedia
   https://en.wikipedia.org/wiki/Complex_Wishart_Distribution