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**WWTL@WHL** This involves a bit of distribution theory but I think the solution involves more linear algebra than statistics so I'm posting here.

**Question:**

A is an idempotent matrix. If $\displaystyle X \sim N (0, I_k )$ (where X is a n dimensional vector). The rank of A is m. Show that $\displaystyle X'AX \sim \chi ^2_m $. (Hint: Use the fact that A's eigenvalues are either 0 or 1)

**My attempt:**

I'm thinking I have to diagonalise A?

So let A = SDS' where D is a diagonal matrix with A's eigenvalues on its diagonal.

So $\displaystyle X'AX = X'SD^{\frac{1}{2}}D^{\frac{1}{2}}S'X $

Let $\displaystyle T = D^{\frac{1}{2}}S'X $

Then E(T) = 0 and $\displaystyle Var(T) = D^{\frac{1}{2}}S'SD^{\frac{1}{2}} $ ??

I think I'm not going anywhere with this.

Please help!