Matrix: Show that it is Symmetric and Idempotent

[Tracker]

X is a n*k matrix, k < n

X is of full rank k (full column rank)

X'X is of full rank and therefore invertible

 

[math] P_x = X(X'X)^{-1}X'[/math]

 

Show that [math]P_x[/math] is symmetric and idempotent.

 

I figured out how to show it is idempotent.

 

Here is my attempt to show it is symmetric:

 

[math] (P_x)' = (X(X'X)^{-1}X')' = X'[(X'X)^{-1}]'X = X'(XX')^{-1}X[/math]

 

 

Maybe someone has a sheet for all the algebra rules for linear algebra that would be helpful?

Edited January 9, 2012 by Tracker