Let matrix $P_j$ denote the $m \times m$ orthogonal projector of rank $m-(j-1)$ that projects $\Re^m $ orthogonally onto the space orthogonal to span ($q_1 .......,q _{j-1}$), where $q_n= \frac {P_n a_n}{||P_n a_n||_2}$ and $a_j$ are the column vectors of an $m \times n$ full rank matrix $A$.
Im trying to understand why the following is true:
$P_j= P_{\perp q_{j-1}} P_{\perp q_{j-2}} \cdot \cdot \cdot\cdot P_{\perp q_{2}} P_{\perp q_{1}}$
Where $P_{\perp q}$ denotes the rank $(m-1)$ orthogonal projector onto the space orthogonal to $q \in \Re^m$.
Just by watching the formula i get the feeling that it is obvious but i just cant prove/clarify it to myself. Any help is appreciated! Thanks