Let $R$ be a real orthogonal matrix,
$$RR^T = I$$
and let $\Omega$ be a real skew-symmetric matrix,
$$\Omega^T = -\Omega$$
Please show (or disprove, although I'm pretty sure it's true) that,
$$ R \Omega = \Omega R$$
I.e. prove whether or not orthogonal matrices and skew-symmetric matrices always commute in multiplication.
Is it possible to show using only the defining properties I listed? Or perhaps it might be necessary to also use the fact that skew-symmetric matrices commute with their transposes.
Best Answer
The claim is false. Consider $$ \Omega=\left[ \begin{array}{ccc} 0&1&0\\ -1&0&0\\ 0&0&0 \end{array} \right] $$ and $$ R=\left[ \begin{array}{ccc} 1&0&0\\ 0&0&1\\ 0&1&0 \end{array} \right].$$
What you may have tried are the two by two matrices, which the commutativity holds except possibly when the orthogonal matrix has determinant $-1$.