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The Hermite Normal Form (HNF) is a canonical representation of matrices overany principal ideal domain. Over the integers, the distribution of the HNFs ofrandomly looking matrices is far from uniform. The aim of this article is topresent an explicit computation of this distribution together with someapplications. More precisely, for integer matrices whose entries are upperbounded in absolute value by a large bound, we compute the asymptotic number ofsuch matrices whose HNF has a prescribed diagonal structure. We apply theseresults to the analysis of some procedures and algorithms whose dynamics dependon the HNF of randomly looking integer matrices.

Natural Density Distribution of Hermite Normal Forms of Integer Matrices