(Generalized Projection) Using the conditions and result of Exercise 4.5, show that

$$P_{X \mid AX}y = \underset{\mu \in Col(X)}{argmin} \ (y - \mu)'A(y - \mu)$$

where

$$P_{X \mid AX} = X(X'AX)^{-1}X'A$$

Show furthermore that $\hat{\mu} = P_{X \mid AX}y = X\hat{\beta}$ implies that

$$\hat{\beta} = (X'AX)^{-1}X'Ay$$