I want to prove the following theorem (I'm not sure but probably it is called Cochran's Theorem) :
Let $X$ follows $n$-variate multivariate normal distribution with mean vector $\mathbf{\mu}$ and covariance matrix $\Sigma$. Also let $A_1,A_2,\dots,A_m$ be symmetric projection matrices of order $n$ such that $rank(A_i)=r_i$ and $\sum_{i=1}^m A_i=I$. Let $U_i=X^TA_iX$. Show that $U_i$'s are independent random variables, respectively distributed as $\chi_{r_i}^2$.
I cannot find the idea to attack the problem. If anyone can give me any lead (even for the simple $m=2$ case, which I may generalize), I'll me much grateful. Thank you.