$u'\Sigma^{-1}u/u'A^{-1}u\sim\chi^2_{n-p}$ for Wishart $A$ with $n-1$ d.f.

If $A\sim W_p(n-1,\Sigma)$ with $n>p$, $|\Sigma|>0$ and U is distributed independently of A (and non-zero), then $\dfrac{u'\Sigma^{-1}u}{u'A^{-1}u}$ follows: