If is the line passing through the origin then, for all and there exists at least one that is, is an eigenvalue of and the corresponding eigenspace is .

Rotations around the origin in and angle have no eigenvalues if and . If we get the identity map (eigenvalue double). If we get (eigenvalue double).My original assumption was that A would not have any eigenvalues because it is a rotation. But someone told me that because A is in R3 and not R2, any point on the line of rotation would be an eigenvalue because it would be the same after the transformation. Could someone confirm this or perhaps clarify it? Do rotation transforms only not contain eigenvalues in R2?