On prescribed characteristic polynomials

Abstract

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 k0-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 Nsuch that the characteristic polynomial of A +Nis exactly q(x). We also show that the restriction k<n −kis necessary in the sense that, when the equality k=n −kholds, not every characteristic polynomial having the same trace as Acan 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