TITLE: Non-commuting orthogonal projections
QUESTION [1 upvotes]: I have some bad intuition. Can anyone provide me with two concrete non-commuting orthogonal projections on a Hilbert space?

REPLY [1 votes]: @Leander's answer is perfect, but it might be interesting to see the concrete matrices associated with the two projections given:
$$
P_S = \pmatrix{1 & 0 \cr 0 & 0},
\qquad
P_T= \pmatrix{{1\over 2} & {1\over 2} \cr {1\over 2} & {1\over 2}},
$$
so that one may verify algebraically that they do not commute:
$$
P_SP_T = \pmatrix{{1\over 2} & {1\over 2} \cr 0 & 0},
$$
while
$$
P_TP_S = \pmatrix{{1\over 2} & 0 \cr {1\over 2} & 0}.
$$
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