
Design matrices problem
I'm a bit rusty on my linear algebra and was given this problem in my advanced linear models class.
Consider the use of two design matrices X and Z for estimating a curve based on the two models y = Xb + e1 and y = Za + e2. Let H = X(X'X)^1*X' denote the hat matrix for the first model. Recall that hat matrices are projection matrices. Consider the case of projecting the design matrix Z onto the column space of X using the transformation Zhat = HZ.
Let bhat = (X'X)^1*X'y denote the least squares estimator of b in the first model. Let a2 denote the least squares estimator of a using the design matrix Zhat. Find an expression for a2 in terms of bhat.
Any help would be greatly appreciated!!

This is how I started:
a2 = (Zhat'*Zhat)^1*Zhat'*y
= ((HZ)'(HZ))^1*(HZ)'*y
= (Z'H'HZ)^1*Z'H'y
= (Z'HZ)^1*Z'Hy
= (Z'HZ)^1*Z'Xb
= (Z'X(X'X)^1*X'Z)^1*Z'Xb
= (X'Z)^1*(X'X)*(Z'X)^1*Z'Xb
= (X'Z)^1*(X'X)b
Can I simplify this further?
Thanks!!