Hermite canonical form and Smith canonical form of a matrix over a principal ideal domain

Abstract

This paper presents computational aspects of Smith and Hermite normal forms for matrices over general principal ideal domains. After reviewing the standard definitions and properties of Hermite and Smith normal forms, we ask when there is an effective algorithm which will find a unique canonical form for all matrices in the same equivalence class. We show that the ability to compute canonical members of an associate class and canonical representatives of a residue class in the underlying ring are necessary and sufficient conditions for an effective procedure to find unique Hermite and Smith forms. We then present a uniform technique for reducing intermediate expression swell during the computation of Smith and Hermite normal forms for matrices over general principal ideal domains. This generalizes previous work by handling non-square matrices which are not of full column rank. We also present examples which show that the heuristic of directly computing modulo the determinant as proposed by some can yield the wrong answer. We instead compute with the matrix augmented by the determinant times the identity matrix. This always yields the correct result and has the desired effect of allowing one to keep all elements which occur during the computation bounded by the determinant.