Eigenvectors are unique up to a scalar If $ A $ is a matrix with eigenvector $ v $ corresponding to the eigenvalue $ \lambda, $ can we prove that $ v $ is unique up to $ \lambda, $ that is if $ v $ and $ v' $ are eigenvectors corresponding to $ \lambda, $ then $ v = Cv' $ for some constant $ C. $

No. It is quite possible for an eigenspace to have more than one dimension. As commenters above pointed out, examples include the identity matrix or the zero matrix.
