Applications of Polynomial Smith Normal form Calculations

An integer Square matrix with a deteminant of + 1 or −1 is called unimodular. Given a (m,m) integer matrix A, there exist unimodular matrices U,K such that S(A)=UAK is a diagonal matrix with positive diagonal elements d 1 ,...,d r (r:=rank(A)) and zero diagonal elements d r+1 ,...,d m . In particular d i divides d i+1 (i=1,...,r−1). This was proved by Smith [21] in 1861 and the matrix S(A) is known as the Smith normal form of A.