Consider the following linear regression model: y = Xβ + u = X1β1 + X2β2 + u
where y and u are n × 1 vectors, X1 and X2 are n × k1 and n × k2 matrices of explanatory variables.
Define the following projections PX1 = X1 (X1^T X1)^(−1)X1^T and MX1 = In − PX1
PX2 = X2 (X2^T X2)−1 X2^T, and MX2 = In − PX2
Prove that: Px1*Px = Px1, and Mx1*Mx = Mx
I understand this intuitively- ie. the expressions make sense, but I cannot for the life of me mathematically prove it. Any help would be v much appreciated.