On prescribed characteristic polynomials

Let F be a field. We show that given any n th degree monic polynomial q ( x ) is an element of F [x] and any matrix A is an element of M-n ( 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. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).