Finding Matrix of improper rotation

Question

I'm struggling with this excercise:

Give a Matrix $A\in O(3)$ which is describing an improper rotation with an angle of $\pi /3$ and axis $(1,1,1)$.

What do I need to do?

Answer 1

You need to find an orthogonal basis whose vectors are transformed by this rotation in an easy manner, so you can write the matrix of this transformation in this basis. (Tip : $(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$ is one of them, why?)

Then, using transition matrices, you compute the matrix of the rotation in the usual basis of R^3

EDIT : A basis would be $(1,1,1)$ and two orthogonal verctors orthogonal to $(1,1,1)$.Then you normalize it by dividing each by its norm.

Example : $((1,1,1), (1,1,-2), (-1, 1, 0))$ gives, once normalized : $$((\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}), (\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{-2}{\sqrt{6}}), (\frac{-1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0))$$

Let A be the rotation matrix as stated in your comment. Let P be the matrix of the good basis (putting the vectors in columns). So the matrix of the rotation in the usual basis is $PAP^{-1}$, which is actually $PA^{t}P$ because P is an orthogonal matrix. This is a standard change of basis.