Eigenvectors with distinct eigenvalues are linearly independant, but what about the case where eigenvalues are repeated.
An example is where we have matrix A = $$ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} $$
with eigenvalues $λ$ = 1 which is repeated twice so spectrum A is 1(2).
And we find that the associated augments matrix is $$ \begin{matrix} 0 & 0 \\ 0 & 0 \\ \end{matrix} $$ And because neither component of the vector is contrained, the eigenvectors is of the form [u v]^T. We observe that [u v]^T = u[1 0]^T + v[0 1]^T. Where {[1 0]^T + [0 1]^T} is linearly independant. So from here how does this actually show that eigenvectors with repeated eigenvalues are not linearly dependant or otherwise?