When is a matrix with repeated eigenvalues diagonalizable?

Question

Let $A_{n\times n}$ have $0<k \leq n$ repeated eigenvalues. Is there any condition that I can impose on $A$, that guarantees that it is diagonalizable? That is, I'm looking for conditions on $A$ which are broad enough to allow repeated eigenvalues, but constrained enough to ensure that there are $k$ linearly independent eigenvectors corresponding to the $k$ repeated eigenvalues.