How are the conditions for "diagonalizability" and "upper-triangularizability" of a linear operator different?

Question

My understanding is that a linear operator is basically diagonalizable if it has as many eigenvectors as its dimension. But when can a linear operator be turned into a upper triangular matrix?

Answer 1

Hint: A sufficient but not necessary condition is that if the minimal polynomial splits into distinct linear factors then the operator is diagonalizable and if the minimal polynomial splits into linear factors not necessarily distinct then the operator is upper triangularizable.