TITLE: Repeated eigenvalues: How to check if eigenvectors are linearly independent or not?
QUESTION [5 upvotes]: I have two related questions. My question is hypothetical, i.e. not from an actual physical problem. If I give you a matrix and tell you that it has a repeated eigenvalue, can you say anything about whether the eigenvectors of that repeated eigenvalue are linearly independent or not? Is there a general procedure to check? I don't know the elements of the matrix itself so I can't work out the eigenvectors in the usual way.
On that note, can someone provide me an example of a matrix with a repeated eigenvalue but where the eigenvectors are not linearly independent? 
Assume the field is complex numbers.

REPLY [3 votes]: Let $A$ be an $n \times n$ matrix with distinct eigenvalues $L_1, ..., L_k$ (note that some of these may have multiplicity > $1$, i.e, be repeated). Let $B_i$ be a basis for the eigenspace of $L_i$ and let $B$ be the union of $B_1, ..., B_k$. Then elements of $B$ are linearly independent.