On prescribed characteristic polynomials

Let F be a field. We show that given any nth degree monic polynomial q(x)∈F[x] and any matrix A∈Mn(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.