Linear Algebra – Similarity of Matrices and 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