The distribution of xtΣx

Question

For the multivariate gaussian distribution $x$, what is the distribution of $x^T\Sigma^{-1}x$?

Answer 1

Hint: If $X \sim N(\mu,\Sigma)$, then $X=\mu + \Sigma^{1/2} Z$ where $Z \sim N(0,I)$.

If $X \sim N(0,\Sigma)$, then $$X^\top \Sigma^{-1} X = Z^\top \Sigma^{1/2}\Sigma^{-1} \Sigma^{1/2} Z = Z^\top Z\sim \chi^2_n.$$