I saw a bunch of examples of Jordan Canonical Form and how it is related to the minimal polynomial. I have noticed the following patterns:
Let $A$ be a matrix and $\lambda_1,\dots,\lambda_k$ be the eigenvalues of $A$.
(1). The number of Jordan blocks corresponding to the eigenvalue $\lambda_j$ is the dimension of the eigenspace of $\lambda_j$.
(2). The power of $(x-\lambda_j)$ in the minimal polynomial of $A$ is the size of the largest Jordan block corresponding to $\lambda_j$.
Are (1) and (2) true in general?