On prescribed characteristic polynomials

Let $F$ be a field. We show that given any nth degree monic polynomial $q\left(x\right)\in F\left[x\right]$ and any matrix $A\in M_n\left(F\right)$ whose trace coincides with the trace of $q\left(x\right)$ 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 $q\left(x\right)$. 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.