Are two matrices similar iff they have the same Jordan Canonical form?

Question

Are two matrices similar if and only if they have the same Jordan Canonical form?

Does the Jordan form have to have ordered eigenvalues?

For example, if $\lambda_1$ and $\lambda_2$ are eigenvalues of $A$, are $\begin{pmatrix}\lambda_1&0\\0&\lambda_2\end{pmatrix}$ and $\begin{pmatrix}\lambda_2&0\\0&\lambda_1\end{pmatrix}$ both Jordan forms of $A$?

Answer 1

• Up to arbitrary ordering of Jordan blocks, yes
• No
• Yes

Answer 2

The eigenvalues need not be ordered. You can conjugate a diagonal matrix by a permutation matrix to put them in any order.