TITLE: Integer matrix with particular Jordan's form
QUESTION [7 upvotes]: For teaching purposes I would like to find integer matrices with a particular Jordan's form. Is there some kind of technique to find nice examples? For example for $$\begin{pmatrix}1&1&0\\0&1&0\\0&0&1\end{pmatrix}.$$

REPLY [4 votes]: As J.M. pointed out in his comment, for any Jordan form $J$ and unimodular matrix $U$, the matrix product $UJU^{-1}$ will do.  In particular, you can get sufficient varieties by building $P$ from a Pascal matrix. For example, the upper triangular order-3 Pascal matrix is
$$
P=\begin{pmatrix}1&1&1\\0&1&2\\0&0&1\end{pmatrix}.
$$
You can take $U$ as any product of $P,\ P^T,\ P^{-1},\ (P^T)^{-1}$, diagonal matrices with diagonal entries $=\pm1$ and permutation matrices. To illustrate, let $J$ be the Jordan form in your example.  Then
$$
\begin{eqnarray}
&&U=PP^TP \Rightarrow UJU^{-1} =\begin{pmatrix}-8&24&-45\\-9&25&-45\\-3&8&-14\end{pmatrix},\\
&&U=P^2\begin{pmatrix}0&-1&0\\0&0&1\\-1&0&0\end{pmatrix}(P^T)^{-1} \Rightarrow UJU^{-1} =\begin{pmatrix}2&-2&5\\3&-5&15\\1&-2&6\end{pmatrix}.
\end{eqnarray}
$$
You can generate a lot of integer matrices with identical Jordan form but very different appearances using this method.