# Idempotent matrix

• Oct 17th 2009, 06:02 PM
WWTL@WHL
Idempotent matrix
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 $X \sim N (0, I_k )$ (where X is a n dimensional vector). The rank of A is m. Show that $X'AX \sim \chi ^2_m$. (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 $X'AX = X'SD^{\frac{1}{2}}D^{\frac{1}{2}}S'X$

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

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

I think I'm not going anywhere with this.

• Oct 17th 2009, 08:52 PM
tonio
Quote:

Originally Posted by 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 $X \sim N (0, I_k )$ (where X is a n dimensional vector). The rank of A is m. Show that $X'AX \sim \chi ^2_m$. (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 $X'AX = X'SD^{\frac{1}{2}}D^{\frac{1}{2}}S'X$

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

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

I think I'm not going anywhere with this.