Eigenvector of a matrix in $S{O}_3(\mathbb{R})$

Abstract

Let $A=[a_{ij}]\in O_3(\mathbb{R})$. We give several different proofs of the fact that the vector $$V:={\left[\begin{array}{ccc}\frac{1}{a_{23}+a_{32}}& \frac{1}{a_{13}+a_{31}}& \frac{1}{a_{12}+a_{21}}\end{array}\right]}^T,$$ if it exists, is an eigenvector of $A$ corresponding to the eigenvalue $1$.

15 pages