Let $A$ be an $n \times n$ matrix with entries in $\mathbb{C}$. Suppose that the characteristic polynomial of $A$ is the same as the minimal polynomial of $A$ such that $$p\_{A}(t) = m\_{A}(t) = (t - \lambda)^{n}$$
What can be said about the number of linear independent eigenvectors for the eigenvalue $\lambda$?