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 $\ P_UP_V = P_VP_U$?
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1https://math.stackexchange.com/q/536664/4280 seems to answer your question. – Henno Brandsma Aug 27 '17 at 22:09
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3I don't think they answer quite the same question. That question linked above answers this question when we know that $P_U P_V = P_{U\cap V}$, but we don't necessarily have that unless we have the answer below. – occassional user Dec 07 '17 at 16:58
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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.
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