Physically, unitary matrices of the same dimension describes a rigid motion, so it feels like the order of the rigid motion doesn't really matter? So do unitary matrices commute?
2026-03-26 09:21:47.1774516907
Do unitary matrices commute?
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Rotations in $\mathbb R^3$ do not commute. Consider $$A=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \\ \end{pmatrix}$$
and $$B=\begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$$
$A$ is a rotation about the $x$ axis and $B$ is a rotation about the $z$ axis. I will leave it to you to verify that each has determinant $1$ and $AB\ne BA$.