Here is the question to which I am referring, along with its solution.:

Question #4:

https://www.docdroid.net/CtqTsdX/e2.pdf

Question #4's Solution:

https://www.docdroid.net/CtqTsdX/e2.pdf#page=3

The solution seems to try to get the reader to think about the significance of the computations sought, but it doesn't seem to specify what that significance is, and I'd like to know, so I asked someone, and I was told that the eigenvalue 1 is a generator of the rotation axis, since RX = 1X = X means that the point X is not moved by the rotation R and that the other two eigenvalues don't have such an obvious representation, and that that just generate the plane in which the rotation happens.

Unfortunately, I'm still not clear on the matter.

I do understand how the point X is not moved by rotation R, but I don't get how that is a "generator of the rotation axis" (other than the fact that rotating in a plane perpendicular to a certain axis is a rotation about that axis). Also, about the other two eigenvalues, I understand that there generally are rotations happening in a plane (such as rotating about the z axis being equivalent to rotating on the xy plane), but I don't see how the eigenvalues generate planes (in which rotations happen).

Could someone please help further clarify this in my mind?

Any input would be GREATLY appreciated!