Let $X \sim \mathcal{W}_p(V,\nu)$ follow a central Wishart distribution with scale matrix $V$ and $\nu$ degrees of freedom. Its p.d.f. is given by: $$ \frac{|\mathbf{X}|^{(\nu-p-1)/2} e^{-\operatorname{tr}(\mathbf{V}^{-1}\mathbf{X})/2}}{2^\frac{\nu p}{2}|{\mathbf V}|^{\nu/2}\Gamma_p(\frac \nu 2)} $$ The variance can be described by the $n^2 \times n^2$ matrix $$ Cov(vec(X))=\frac{1}{\nu} \left(I_{n^2}+K_{nn}\right)\left(V \otimes V\right), $$ where $K_{nn}$ is the commutation matrix defined by $Kvec(A)=vec(A')$.

How can we calculate this covariance matrix? Is there a general way to do it for matrix valued distributions?



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