Fillmore's theorem for integer matrices

Fillmore Theorem says that if A is a nonscalar matrix of order n over a field F and γ1,…,γn∈F are such that γ1+⋯+γn=trA, then there is a matrix B similar to A with diagonal (γ1,…,γn). Fillmore's proof works by induction on the size of A and implicitly provides an algorithm to construct B. We develop an explicit and extremely simple algorithm that finish in two steps (two similarities), and with its help we extend Fillmore Theorem to integers (if A is integer then we can require B to be integer).