On the product of a rotation matrix and a skew symmetric matrix

Question

Given $R \in SO(3)$ such that $tr(R) = -1$ ($tr(.)$ denotes the trace operation) and $b \in \mathbb{R}^3$ can it be proven that $b_{\times}R$ is symmetric? ($b_{\times} \in so(3) , b_{\times}a = b \times a$ for any $a \in \mathbb{R}^3$)

Answer 1

Would $$R=\pmatrix{1&0&0\\0&-1&0\\0&0&-1}$$ and $$b_\times=\pmatrix{0&1&1\\-1&0&1\\-1&-1&0}$$ satisfy your conditions? $b_\times R$ is not symmetric.

Answer 2

You cannot prove that because it is not true. The spectrum of a rotation matrix $R\in SO(3,\mathbb R)$ for an angle $\theta$ is $\{1,e^{i\theta},e^{-i\theta}\}$. If $\operatorname{tr}(R)=-1$, the spectrum must be equal to $\{1,-1,-1\}$. Therefore, by a change of orthonormal basis, we may assume that $R=\operatorname{diag}(1,-1,-1)$. By inspecting the entries of $b_\times R$ directly, we see that $b_\times R$ is symmetric if and only if $b$ is in the form of $(0,y,z)^T$, i.e. iff $b$ lies on the plane of rotation of $R$.