when do orthogonal projection $\ P_UP_V = P_VP_U$

Question

U, V are subspaces of a finite dimensional vector space W, Let $\ P_U$ and $\ P_V$ be the orthogonal projections onto U and V respectively. When is it true that $\ P_UP_V = P_VP_U$?

Answer 1

This is true if and only if $U$ are $V$ are "perpendicular". By perpendicular, we mean the following:

Consider $U\cap V$ and its orthogonal complement in $U$, respectively $V$, denoted by $U_1$, $V_1$. We require that $U_1$ orthogonal to $V_1$ in the usual sense.