On prescribed characteristic polynomials

Let F be a field. We show that given any nth degree monic polynomial q (x) Fx and any matrix Aₙ (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.