I want to prove Euler's rotation theorem:
In three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point.
I have just finished studying the matrix proof of Euler's rotation theorem as stated in the wikipedia page here.
The author just takes an arbitrary $3 \times 3$ rotation matrix $R$ and proves that $\lambda = 1$ is eigenvalue for this matrix. I do not understand why this proves that such rigid body displacement is rotation by some axis. For me it only shows rotation matrix properties.
Some point is fixed, so take this as the origin. Write the rotation as $x\mapsto Rx$. Since $R$'s eigenvalues are $\pm1$, an arbitrary point is of the form $x_++x_-$ with $Rx_\pm=\pm x_\pm$. Then $Rx=x_+-x_-$, so $x$ is fixed iff $x_-=0$, i.e. iff $x$ is on the axis through $O$ parallel to an eigenvalue-$1$ vector.