# Idempotent matrix

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• October 17th 2009, 05:02 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.

Please help!
• October 17th 2009, 07:52 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.

Please help!

What is N(0,I_k)? What does " X ~ N(0, I_k) " mean? What is X_m? A characteristic fucntion of some set?
Later you write about E(T) and Var(T)...what are these??

Perhaps this is related to probability or something, but this section is about algebra, and the above looks chinese to me.

Tonio