Axis of rotation

1. Feb 17, 2013

1. The problem statement, all variables and given/known data

consider the following rotation matrix:

0 0 1
1 0 0
0 1 0

Find the axis of rotation.

2. Relevant equations

3. The attempt at a solution

I know the following:

Ω|1> = |2>
Ω|2> = |3>
Ω|3> = |1>

where Ω is an operator.

It is a cyclic permutation. What do not understand is how the rotation axis is |1>+|2>+|3>/(3^0.5)

2. Feb 17, 2013

vela

Staff Emeritus
Hint: Try finding the eigenvectors of Ω.

3. Feb 17, 2013

jfgobin

A rotation along an axis doesn't change the axis itself.

Besides the obvious $\left ( 0,0,0 \right )$, do you see another vector that would satisfy

$\begin{pmatrix} 0&0&1\\ 1&0&0\\ 0&1&0 \end{pmatrix} \cdot \vec{v} = \lambda.\vec{v}$​

As Vela suggested, this will give you the eigenvalues and eigenvectors or your matrix, the latter being the vectors whose magnitude is changed by the linear application.

4. Feb 17, 2013

I get λ=1 and indeed I get the rotation vector as stated in the question, but tell me why would an eigenvalue of the rotation matrix give me the axis of rotation? Is it because eigen values tell us how much a system is dependent on the variables in the system. So in this particular case I have 1,2,3 as the basis and the λ would give me the dependence of the rotation matrix on the basis! right?

5. Feb 17, 2013

jfgobin

$\lambda=1$ means that the corresponding eigenvector is left unchanged: it is not rotated nor does its magnitude change.