# Eigenvalues and eigenspaces of orthogonal (or rotation) matrices

Given an orthogonal transformation of finite order, e.g.

```
Matrix([[0,0,0,-1],[1,0,0,-1],[0,1,0,-1],[0,0,1,-1]])
```

Its eigenvalues are going to be of the form

```
exp(I*pi/5),exp(2*I*pi/5),...,exp(2*I*pi*m),...
```

corresponding to a splitting of the matrix into rotation (and reflection) matrices. I'd like to extract these fractions `m`

(mod ZZ) and study the corresponding (real rotation and reflection) eigenspaces.

My impression is that Sage isn't suitable for doing this directly, but that I should use e.g. the Maxima or Mathematica interface? Any suggestions for the most suitable method?

Well... here is a start: