Let $K$ be a field, $X$ an indeterminate, and $V$ a finite dimensional $K[X]$-module.
Recall that the multiset of elementary divisors of $V$ is the multiset
$$f_1^{n(1,1)},\dots,f_1^{n(1,k(1))},$$
$$\vdots$$
$$f_r^{n(r,1)},\dots,f_r^{n(r,k(r))},$$
defined by the following conditions: the $f_i$ are distinct monic irreducible polynomials, the $n(i,j)$ satisfy $$n(i,1)\ge\cdots\ge n(i,k(i)),$$
and we have
$$V\simeq\bigoplus_{i,j}\frac{K[X]}{\big(f_i^{n(i,j)}\big)}\quad.$$
The characteristic polynomial is the product of the elementary divisors, whereas the minimal polynomial is the product of the $f_i^{n(i,1)}$.
If $W$ is also a finite dimensional $K[X]$-module, then the multiset of elementary divisors of $V\oplus W$ is the "disjoint union" (in the obvious sense) of the multisets of elementary divisors of $V$ and $W$.
Some diagonal entries of Nir's matrix are equal to 1, and the others are
$$X,\quad X+2,\quad X^2,\quad X+2,\quad (X+2i)(X-2i)X,\quad (X+2)^2.$$
The multiset of elementary divisors is thus
$$\begin{matrix}
(X+2)^2,&X+2,&X+2,\\ \\
X^2,&X,&X,\\ \\
X+2i,&&\\ \\
X-2i.&&
\end{matrix}$$
The characteristic polynomial is $$(X+2)^4X^4(X+2i)(X-2i).$$ The minimal polynomial is $$(X+2)^2X^2(X+2i)(X-2i).$$
The Jordan blocks $J(\lambda,n)$ are
$$\begin{matrix}
J(-2,2),&J(-2,1),&J(-2,1),\\ \\
J(0,2),&J(0,1),&J(0,1),\\ \\
J(-2i,1),&&\\ \\
J(2i,1).&&
\end{matrix}$$
Let's compute the invariant factors (even if it wasn't required). In the array of elementary divisors, there are four blank entries. We put a $1$ in each of them:
$$\begin{matrix}
(X+2)^2,&X+2,&X+2,\\ \\
X^2,&X,&X,\\ \\
X+2i,&1,&1,\\ \\
X-2i,&1,&1.
\end{matrix}$$
We get the first invariant factor by multiplying together the polynomials in the first row, and so on. Thus the invariant factors are
$$(X+2)^2X^2(X+2i)(X-2i),\quad (X+2)X,\quad (X+2)X.$$
[Reference: Bourbaki, Algèbre, VII.4.8.]