What Is the Axis of Rotation for This Matrix?

[rsaad]

Homework Statement

consider the following rotation matrix:

0 0 1
1 0 0
0 1 0

Find the axis of rotation.

Homework Equations

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)

 

[vela]

Hint: Try finding the eigenvectors of Ω.

 

[jfgobin]

Hello Rsaad,

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

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

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.

 

[rsaad]

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?

 

[jfgobin]

Rsaad,

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

Have you read the link to the wiki page I posted?

 

[rsaad]

Yes, I understand that and yes I had a look at that page.