I'm having difficulty with this question from my Linear Algebra course,

Question details

I don't know how to answer the following,

Find the axis of rotation of the composition L1 ◦ L2

  • $\begingroup$ Have you learned about eigenvalues and eigenvectors yet? $\endgroup$ Apr 25 '18 at 12:44

What you're looking for is the eigenvector associated with $\lambda = 1$ for the composition $L_1 \circ L_2$. In other words, you want a non-zero vector $v$ which will solve the equation $L_1(L_2(v)) = v$. This system can be routinely solved using the matrices for the transformations $L_2$ and $L_1$.

To think about it in an intuitive way, we want a vector for which the rotation about $e_3$ moves $v$ to $L_2(v)$, and the rotation about $e_1$ moves $L_2v$ back to $v$.

As it turns out, the correct vector to choose is $$ v = (1,1,-1) $$ (or any non-zero multiple of this vector). Verify that $$ L_2(v) = (1,-1,-1)\\ L_1(L_2(v)) = L_1(1,-1,-1) = (1,1,-1) $$

  • $\begingroup$ but axis of rotation has also a direction (two possibilities) , how do you solve this problem from eigenvectors ? $\endgroup$
    – Widawensen
    Apr 25 '18 at 12:57
  • $\begingroup$ @Widawensen didn't consider that. Also, I now realize that I missed that $L_2$ is clockwise. $\endgroup$ Apr 25 '18 at 12:59
  • $\begingroup$ @Widawensen actually, I'm betting that the book posing this question doesn't care about the axis direction; note that it refers to a clockwise rotation about the $e_3$-axis rather than a counterclockwise rotation about the $-e_3$-axis, which would be the correct "direction" for the axis of that rotation. $\endgroup$ Apr 25 '18 at 13:05
  • $\begingroup$ If direction is not important then eigenvector is sufficient. However by changing the direction of the axis we are substancially changing the final result of rotation as convention for +/- angle is valid only with directed axis ( right hand rule) .. $\endgroup$
    – Widawensen
    Apr 25 '18 at 13:08
  • $\begingroup$ @Widawensen true, but note that they're not asking for the angle of the rotation $\endgroup$ Apr 25 '18 at 13:09

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