  We study representations of integral vectors in a number system with a matrix
base $M$ and vector digits. We focus on the case when $M$ is similar to $J_n$,
the Jordan block of $1$ of size $n$. If $M=J_2$, we classify digit sets of size
2 allowing representation of the whole $\mathbb{Z}^2$. For $J_n$ with $n\geq
3$, it is shown that three digits suffice to represent all of $\mathbb{Z}^n$.
For bases similar to $J_n$, at most $n$ digits are required, with the exception
of $n=1$. Moreover, the language of strings representing the zero vector with
$M=J_2$ and the digits $(0,\pm 1)^T$ is shown not to be context-free, but to be
recognizable by a Turing machine with logarithmic memory.
