I am trying to prove the following result ($x$ ~ $N_p(m,\Sigma)$): $$\mathrm{var}[x^\top A x] = 2\mathrm{Tr}(A\Sigma A\Sigma) + 4 m^\top A \Sigma A m $$ I have completed most of it except that $E[yy^TAyy^T]=2\Sigma A\Sigma$ where $y$~$N_p(0,\Sigma)$. Any hints for this? I know $yy^T$ follows Wishart$_p(\Sigma, p)$ but not sure if that will help. I only know mean of Wishart will be $\Sigma$. Dont know anything about Variance of the Wishart
Asked
Active
Viewed 87 times
0

Does [this](https://math.stackexchange.com/a/442916/321264) help? We don't need to know the exact distribution of the quadratic form. – StubbornAtom Mar 01 '20 at 17:23

@StubbornAtom Yes, it does. I didn't explicitly compute what I asked in question but was able to find the distribution of $x^TAx$ and then the result. Thank you – Anvit Mar 01 '20 at 18:04

1You can also find an answer [here](https://stats.stackexchange.com/q/303466/119261). – StubbornAtom Jul 13 '20 at 17:54

1@StubbornAtom That helps, thanks! – Anvit Jul 13 '20 at 18:44