Projection matrix multiplied by projection matrix.

Question

Let $X_1,X_2,...X_n$ be $n$ x $1$ vectors. Let $X_j$ be the matrix composed by the vectors $X_1,...,X_j$ and $X_i$ be the matrix composed by the vectors $X_1,...,X_i$, where $i<j$. The projection matrices and are the matrices associated with the matrices $X_j$ and $X_i$.

How can be proved that $PX_iPX_j=PX_i$?

Answer 1

Your notation is not too lucky. Let's rather call the matrices $A_i=[X_1,\dots,X_i]$ and the orthogonal projections $P_i$ which project on the range of $A_i$, i.e. on ${\rm span}(X_1,\dots,X_i)$.

Now you only need to prove that $P_UP_V=P_U$ whenever $U\subseteq V$ subspaces, where $P_U$ denotes the orthogonal projection in $U$.

Let $x$ be any vector and let $v=P_Vx\in V$, so that $x=v+w$ with $w\perp V$. Then $w\perp U$ as well, and thus $P_Uw=0$, and $P_UP_Vx=P_Uv=P_U(v+w)=P_Ux$.