This involves a bit of distribution theory but I think the solution involves more linear algebra than statistics so I'm posting here.
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)
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.
Then E(T) = 0 and ??
I think I'm not going anywhere with this.