# Deriving expectation involving Wishart distributions $E[\bf{A(A'WA)^-A'W}]=\bf{A(A'\Sigma A)^-A'\Sigma}$

I have a problem deriving two expectation involving Wishart distributions with mean zero.

Let $\bf{W} \sim {W_p}({\bf{\Sigma }},$$n\bf{)} and \bf{A}$$: p\times q$. Prove that

$E[\bf{A(A'WA)^-A'W}]=\bf{A(A'\Sigma A)^-A'\Sigma}$

and

$E[\bf{A(A'W^{-1}A)^-A'W^{-1}}]=\bf{A(A'\Sigma^{-1} A)^-A'\Sigma^{-1}}$.

I have noticed that both expressions are idempotent matrices but I dont know if that is of any help. I have also found the following Theorem that I thought could be used:

• $\bf{A(A'W^{-1}A)^-A'}\sim \bf{W}_p(A(A'\Sigma^{-1}A)^-A'$$,n-p+r(\bf{A}))$
• $\bf{A(A'W^{-1}A)^-A'}$ and $\bf{W-\bf{A(A'W^{-1}A)^-A'}}$ are independent
• $\bf{A(A'W^{-1}A)^-A'}$ and $\bf{I-\bf{A(A'W^{-1}A)^-A'}}$

Note that $\bf{B}^-$ stands for the general inverse of $\bf{B}$.

Any help is appreciated.

• Anyone have an idea? – Syltherien Sep 21 '15 at 7:48