Sorry for boring you my friends. I am haunted by a demonstration in the book. Here is the link: http://robotics.caltech.edu/~jwb/courses/ME115/handouts/rotation.pdf
The question is mainly about the complex eigenvalue of a 3 dimensional rotation matrix. In order to introduce the angle of rotation into the characteristic equation. On page 3,the Equation 9, the author directly use the formula of the trace of the rotation matrix values as $1+2\cos(\phi)$ without any demonstration. I would like to know the appropriate way to introduce the angle of rotation.
Thank you in advance for taking a look.
look at the second degree equation in the formula (8): $$ \lambda^2-\lambda(a_{11}+a_{22}+a_{33}-1)+1=0 $$ The solutions are: $\lambda_3=\overline{\lambda_2}$ with $\lambda_3\lambda_2=\overline{\lambda_2}\lambda_2=1$, so $|\lambda_2|=|\lambda_3|=1$ and the solutions, if they are complex numbers, have the form: $$ \lambda_{2,3}=\cos \phi \pm i \sin \phi $$ and, since the sum is: $$ \lambda_2+\lambda_3=2 \cos \phi=a_{11}+a_{22}+a_{33}-1 $$ we have the result in (9).