Matrices with Integer Entries: The Smith Normal Form

Abstract

Let A denote an s×t matrix with integer entries. By applying elementary row and column operations A is ‘reduced’ to a diagonal matrix D=diag (d 1,d 2,d 3,…) with non-negative integer entries such that d 1 is a divisor of d 2,d 2 is a divisor of d 3 and so on. The reduction process is a matrix version of the Euclidean algorithm for calculating gcds and is expressed by PA=DQ where P and Q are invertible matrices over ℤ arising from the row and column operations used. The ring ℤ of integers is a principal ideal domain and the Cauchy–Binet theorem on determinants is proved. For each A the matrix D is shown to be unique and is called the Smith normal form of A after H.J.S. Smith (1861). The notation D=S(A) is introduced.KeywordsSmith Normal FormInteger EntriesBinet-Cauchy TheoremEuclidean AlgorithmPrincipal Ideal Domain (PID)These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.