Orthogonal projection and subspaces proof? Let's let $M$ be a subspace of $\mathbb{R}^n$ and let $N$ be a subspace of $M$. 
Let $m$ and $n$ denote the orthogonal projection matrices onto $M$ and $N$. Show that $mn = nm = n$.
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I'm thinking about inner products of $n$ and $m$, but they would only be zero in the area in which $N$ is a subspace of $M$. I'm a bit lost. Any advice/pointers?
 A: Denoting matrices with small letters is confusing. I'll denote projections on $M$ and $N$ as $P_M$ and $P_N$ respectively. I will also denote $V = \mathbb{R}^n$, as this proof is good for any finite inner product space.
Note that we can write $V$ as a direct sum of three orthogonal spaces (see Wikipedia article on orthogonal complements):
$$V = M \oplus M^\perp = (V \cap M) \oplus M^\perp = ((N \oplus N^\perp) \cap M) \oplus M^\perp = N \oplus (N^\perp \cap M) \oplus M^\perp.$$
Let $v = x + y + z$, be any vector in $V$, where $x \in N$, $y \in N^\perp \cap M$, $z \in M^\perp$. Recall (Lemma 4.8 here) that for each $v \in V$ there exist (unique) $x,y,z$, so we can do this for any $v \in V$. Then
\begin{gather*}
P_N x = x, \quad P_M x = x, \\
P_N y = 0, \quad P_M y = y, \\
P_N z = 0, \quad P_M z = 0,
\end{gather*}
so
\begin{align*}
P_M P_N v &= P_M P_N (x + y + z) = x, \\
P_N P_M v &= P_N P_M (x + y + z) = x, \\
P_N v &= P_N (x + y + z) = x,
\end{align*}
which proves the statement.
A: Presumably the statement was meant to be: Show $mn =nm = n$.
Note that if $P$ is a projection, then $x \in {\cal R} P=$ iff $Px = x$. (One direction is trivial, the other follows from $x = Py = P(Py + (I-P) y) = P(x + (I-P)y) = P x$.)
Suppose $N = {\cal R} n \subset M = {\cal R} m$.
First show $mn=n$. Choose $x$. Then $nx \in {\cal R} n \subset{\cal R} m$, hence $nx = mnx$, from which the result follows. This is true for all projections, orthogonal or not.
Now show $nm = n$. Here we need $m$ and $n$ to be orthogonal. It is straightforward to show that a projection $m$ is orthogonal iff $m$ is self-adjoint.
Then we have $m^*(I-m) = m(I-m) = 0$. Since $mn=n$, we have $n^* m^* = n^*$, and so $n(I-m) = n^*(I-m) = n^* m^* (I-m) = n^* m (I-m) = 0$.
Now we have $nx = n(mx + (I-m)x) = nm x + n(I-m)x = nmx$, which shows $n = nm$.
(The result is true for any closed subsets $M,N$ of a Hilbert space.)
