# Lines of invariance question

Just need a couple of things confirmed for me guys.

Firstly, lines of invariance are always real eigenvectors right?

Secondly, how is this line of invariance related to rotational matrices? My line of invariance happens to be the same axis. Finally, how is the angle of the rotation calculated?

Cheers.

## Answers and Replies

HallsofIvy
Certainly, a rotation around a given axis leaves that axis invariant- the axis is a "line of invariance". As for "how is the angle of the rotation calculated?", that depends on what information you are given. If you are given the rotation matrix, find the eigenvalues. One, with eigenvector in the direction of the axis of rotation, will be 1, the others will be complex conjugates of the form $e^{i\theta}$, where &theta; is the angle of rotation.