Find the axis of rotation from the rotation matrix.

Question

This is a problem from the book "Mathematical Methods in the Physical Sciences" Third Edition by author Mary L. Boas. on page 129, Example 5, just in case any of you are familiar with it. So I actually know what the correct answers are. Its how to get the answers is what I so desperately need to know.

$$G=\ \begin{pmatrix} 0 & 0 & 1\\ 0 & -1 & 0\\ 1 & 0 & 0 \end{pmatrix}$$

$$K=\ \begin{pmatrix} 0 & 0 & 1\\ -1 & 0 & 0\\ 0 & -1 & 0 \end{pmatrix}$$

Find the axis of rotation for the rotation matrices $G$ and $K$.

I know that many of you can do this by "inspection". But I don't understand what that is or how it works. The book tells me I can solve the equations $Gr=r$ & $Kr=r$ to get the axis of rotations since $r$ is some vector unchanged by the transformation. I haven't reached the Eigenvector section of the book yet so if someone would kindly show me all the working for solving these equations I would be most grateful. As according to Mary Boas I don't need to know about Eigenvectors to solve this problem.

Many thanks to any response,

regards, BLAZE

Answer 1

Note that axis consists of vectors that remain unmoved. That is a $v$ satisfying $G v = v$. Or, $Gv - Iv=0$ where $I$ is the $3\times3$ identity matrix. Or $(G-I)v=0$. So solve the homogeneous equations given by the matrix $G-I$ and get vectors in the axis.

Answer 2

Hint: The column vectors encode where each of the standard basis vectors are sent.

That is, the first column is the vector that $\langle 1, 0, 0 \rangle$ maps to, and so forth.