Distinct eigenvalues of matrix

Question

While studing Quantum Physics I've came to the conclusion that it would be useful to find an equivalent (or at least a sufficient) condition for a matrix (over $\mathbb{C}$) to have distinct eigenvalues.

Clearly I'm not looking for conditions that are too close to the definition (such as that the minimal polynomial is separable).

Can anyone think of any such condition?

If so, can it be generlized to any linear operator? (not necessaraly over a finitely generated vector space)

Answer 1

EDIT. In the formula, I forgot the $\det$.

$A\in M_n(\mathbb{C})$ has $n$ distinct eigenvalues iff $discriminant(\det(A-xI_n),x)\not=0$.

cf. http://en.wikipedia.org/wiki/Discriminant

We may replace $\mathbb{C}$ with any algebraically closed field $K$.