1
$\begingroup$

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)

Improve this question
$\endgroup$
3
  • $\begingroup$ Since $Y$ is defined but never referenced, could there be a typographical error in this question? $\endgroup$
    – whuber ♦
    Commented Feb 4, 2017 at 16:44
  • 1
    $\begingroup$ @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 $a\in \Bbb{R}^p$ $\endgroup$
    – Qwerty
    Commented Feb 4, 2017 at 16:58
  • 1
    $\begingroup$ stats.stackexchange.com/questions/427332/… $\endgroup$
    – StubbornAtom
    Commented Oct 10, 2019 at 18:26

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.