- #1
Calabi_Yau
- 35
- 1
Homework Statement
The probelm is to show, that a rotation matrix R, in a odd-dimensional vector space, leaves unchanged the vectors of at least an one-dimensional subspace.
Homework Equations
This reduces to proving that 1 is an eigenvalue of Rnxn if n is odd. I know that a rotational matrix has determinant = 1 and that it is orthogonal and thus its inverse equals its transpose.
The Attempt at a Solution
I have considered det(R - I) = 0, as there is some v ≠ 0 such that (R - I)v = 0, thus R - I is singular and det(R - I) = 0. Now, what I'm having trouble dealing with, is proving that this only happens for odd-dimensional vector spaces. How do I "insert" the odd-dimension in this problem? Could you give me an hint? I feel like I'm really close but I can't seem to figure it out. :S