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

(where X is a n dimensional vector). The rank of A is m. Show that

. (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

Let

Then E(T) = 0 and

??

I think I'm not going anywhere with this.

Please help!