The Smith normal form

Henry John Stephen Smith (1826-1883) was the Savilian Professor of Geometry at Oxford, and was regarded as one of the best number theorists of his time. His specialties were pure number theory, elliptic functions, and certain aspects of geometry. He shared a prize with H. Minkowski for a paper which ultimately led to the celebrated Hasse-Minkowski theorem on representations of integers by quadratic forms, and much of his research was concerned with quadratic forms in general. He also compiled his now famous Report on the Theory of Numbers, which predated L. E. Dickson’s History of the Theory of Numbers by three-quarters of a century, and includes much of his own original work. The only paper on the Smith normal form (also known as the Smith canonical form) that he wrote [On systems of linear indeterminate equations and congruences, Philos. Trans. Roy. Sot. London CLI:293-326 (1861)] was prompted by his interest in finding the general solution of diophantine systems of linear equations or congruences. Matrix theory per se had not yet developed to any extent, and the numerous applications of Smiths canonical form to this subject were yet to come.