  Let $\mathbb{F}$ be a field. We show that given any $n$th degree monic
polynomial $q(x)\in \mathbb{F}[x]$ and any matrix
$A\in\mathbb{M}_n(\mathbb{F})$ whose trace coincides with the trace of $q(x)$
and consisting in its main diagonal of $k$ 0-blocks of order one, with $k<n-k$,
and an invertible non-derogatory block of order $n-k$, we can construct a
square-zero matrix $N$ such that the characteristic polynomial of $A+N$ is
exactly $q(x)$. We also show that the restriction $k<n-k$ is necessary in the
sense that, when the equality $k=n-k$ holds, not every characteristic
polynomial having the same trace as $A$ can be obtained by adding a square-zero
matrix. Finally, we apply our main result to decompose matrices into the sum of
a square-zero matrix and some other matrix which is either diagonalizable,
invertible, potent or torsion.
