Filmor Theorem for integers

2017/04/26 by Alberto Borobia, Borobia, Alberto
Mathematics · #15A83 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:15A83

paper · pdf · doi:10.48550/arxiv.1704.08037

3 pages

arxiv created 2017/04/26 · arxiv updated 2017/04/27

Abstract

Fillmore Theorem says that if A is a nonscalar matrix of order n over a field \mathbbF and γ₁,…,γ_n∈ \mathbbF are such that γ₁+⋯+γ_n=tr A, then there is a matrix B similar to A with diagonal (γ₁,…,γ_n). Fillmore 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 to B to be integer).

Related

• On Fillmore's theorem over integrally closed domains
• On Fillmore's theorem extended by Borobia
• Fooling-sets and rank in nonzero characteristic (extended abstract)
• Lines of full rank matrices in large subspaces
• On the diagonal of the matrices in a similarity class