How to decide the Jordan normal form of a matrix?

Question

I have the following matrix:

\begin{bmatrix}1&1&0\\-1&3&0\\-1&1&2\end{bmatrix}

My matrix has characteristic polynomial $(X-2)^3$ and minimal polynomial $(X-2)^2$. How do I decide between the two following Jordan normal forms of this matrix?

\begin{bmatrix}2&1&0\\0&2&0\\0&0&2\end{bmatrix}

or

\begin{bmatrix}2&0&0\\0&2&1\\0&0&2\end{bmatrix}

Answer 1

Both of them will do. That is, both of them are a Jordan normal form of your matrix.

Answer 2

They are equivalent Jordan normal form as also the following

$$\begin{bmatrix}2&0&0\\1&2&0\\0&0&2\end{bmatrix} \quad \begin{bmatrix}2&0&0\\0&2&0\\0&1&2\end{bmatrix}$$

which are sometimes also used.

Indeed the Jordan normal form is unique up to a permutation of the blocks.