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Suppose $X\sim N_p(\mu,\Sigma)$ . $A$ is a $p\times p$ symmetric matrix . $Q$ is defined as $Q=X^TAX$. Also $Y$ be defined as $Y=X-\mu$.

Show that $Var(Q)=2\ tr(A\Sigma A\Sigma)+4\mu^TA\Sigma A\mu$

Can someone help ? Should spectral decomposition of $A$ into eigen-values and eigenvectors be of any help?..

(I first tried to break down everything into Double-summation, so as to remove any vector of matrix, but didnt yield anything useful)

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Qwerty
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  • Since $Y$ is defined but never referenced, could there be a typographical error in this question? – whuber Feb 04 '17 at 16:44
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    @whuber No no.. There was another part of the question, which i could solve so i didn't post here. It was to show that $$Cov(a^tY,Y^TAY)=0$$ for any fixed $a\in \Bbb{R}^p$ – Qwerty Feb 04 '17 at 16:58
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    https://stats.stackexchange.com/questions/427332/variance-of-quadratic-form-for-multivariate-normal-distribution – StubbornAtom Oct 10 '19 at 18:26

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