# Eigenvector of a matrix in $SO_3(\mathbb{R})$

## TL;DR

This paper provides multiple proofs that a specific vector, constructed from the entries of a matrix in $O_3(
eal)$, is an eigenvector associated with eigenvalue 1, assuming it exists.

## Contribution

It introduces several different proofs for the eigenvector property of a particular vector in $SO_3(
eal)$ matrices, enhancing understanding of their structure.

## Key findings

- The vector $V$ is an eigenvector of $A$ with eigenvalue 1.
- Multiple proofs confirm the eigenvector property.
- Conditions for the existence of $V$ are discussed.

## Abstract

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