Fillmore's theorem for integer matrices.

Fillmore Theorem says that if A is a nonscalar matrix of order n over a field F and γ 1 , … , γ n ∈ F are such that γ 1 + ⋯ + γ n = tr A , then there is a matrix B similar to A with diagonal ( γ 1 , … , γ n ) . 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). [ABSTRACT FROM AUTHOR]