Verifying understanding of uniqueness of the Jordan form?

Question

I would just like to make sure I understand the uniqueness part correctly. Suppose I know that the Jordan blocks $J_1, ..., J_n$ make up the Jordan form of $A$. Then I can arrange the blocks $J_1, ..., J_n$ in any order on the diagonal and still have a Jordan form of $A$? Furthermore, there exists no other set of Jordan blocks $J_1, ..., J_k$ such that $A$ is similar to a block diagonal matrix constructed from $J_1, ..., J_k$?

Thank you.

Answer 1

Yes the understanding is correct. Jordan form is unique up to permutation of blocks