For (X=left(X_{1}, X_{2} ight)), the generalized inverse of (left(X^{prime} X ight)) is given by (left(X^{prime} X ight)^{-}=left[begin{array}{cc}left(X_{1}^{prime}

For \(X=\left(X_{1}, X_{2}\right)\), the generalized inverse of \(\left(X^{\prime} X\right)\) is given by

\(\left(X^{\prime} X\right)^{-}=\left[\begin{array}{cc}\left(X_{1}^{\prime} X_{1}\right)^{-} & 0 \\ 0 & 0\end{array}\right]+\left[\begin{array}{c}-\left(X_{1}^{\prime} X_{1}\right)^{-} X_{1}^{\prime} X_{2} \\ I\end{array}\right]\left(X_{2}^{\prime} Q_{\left[X_{1}\right]} X_{2}\right)^{-}\left[-X_{2}^{\prime} X_{1}\left(X_{1}^{\prime} X_{1}\right)^{-} I\right]\)

see Davis (2002). Use this result to show that \(P_{[X]}=P_{\left[X_{1}\right]}+\) \(P_{\left[Q_{\left[X_{1}\right]} X_{2}\right]}\). This verifies (9.29).

\[\begin{equation*}
P_{[\Delta]}=P_{\Delta_{1}}+P_{\left[Q_{\left[\Delta_{1}\right]} \Delta_{2}\right]} \tag{9.29}
\end{equation*}\]