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Let $X=(X_1,X_2)$ be a $n \times p$ matrix of rank $p$, where $X_1$ is $n\times p_1$ and $X_2$ is $n\times p_2$. Let $P_1$ be orthogonal projection onto $C(X_1)$ and $P_2$ be orthogonal projection onto $C(X_2)$. Now define $U_2=(I-P_1)X_2$, and let $P_2^*=(U_2)(U_2^T(U_2))(^-)(U_2^T)$. Finally, let $P=P_1+P_2^*$.

Are $P$ and $P_2^*$ orthogonal projections?

Is $U_2$ full rank?

Define $U_1=(I-P_2)X_1$ and $P_1^*=(U_1)(U_1^T(U_1))(^-)(U_1^T)$. Let $Q=P_2+P_1^*$. Does $Q=P$?

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  • $\begingroup$ If $P=P_1+P_2^*$, and both $P_1$ and $P_2^*$ are projections, there is no way $P$ and $P_2^*$ are going to be orthogonal, and $P$ will typically not even be a projection. $\endgroup$ – Thomas Lumley Oct 5 '20 at 6:06
  • $\begingroup$ @Thomas In this setting, $P_1$ and $P_2^{*}$ will be orthogonal projections by construction. $\endgroup$ – whuber Oct 5 '20 at 14:54
  • $\begingroup$ Yes, but he asked if $P$ and $P_2^*$ are orthogonal, where $P=P_1+P_2^*$, and those aren't orthogonal. It's true that $P$ will be a projection, though. $\endgroup$ – Thomas Lumley Oct 5 '20 at 21:10

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