Normal forms — the series
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What a determinant does not determine
Two integer matrices can have the same determinant, the same rank and the same size, and define genuinely different maps. What separates them is a list of integers each dividing the next — computed here twice, once by unimodular elimination and once from the gcds of every minor, which share no algorithm at all.
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Every entry under the determinant
The Hermite form's swell was put down to its transform — the record of how the answer was reached — while the form's own entries stayed bounded. Instrumented separately, the form's intermediates swell exactly as much: 3,371 bits against the transform's 3,366 for a random 48 × 48 matrix whose determinant has 219. Carried out modulo the determinant instead, every entry stays within the determinant's width, the answer is identical on every matrix to n = 80, and it is almost always the identity with the whole determinant in its last pivot and last column. What the bound does not buy is speed: counted in bit operations the modular route does more work until just past n = 72, because its steps multiply numbers the determinant's size where plain elimination multiplies wide numbers by small quotients.
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The swell was one-sided
The Smith form can be computed modulo the determinant, and it was predicted to return the right invariant factors, to stay within twice the determinant's width, and to cost about what the modular Hermite form costs instead of the far larger count of the plain Smith routine. The first three hold. The comparison does not: the plain Smith routine never needed a far larger count. With column operations as well as row operations, its smallest-entry pivot is a unit after a step or two, every entry it holds is a minor of the matrix, and nothing it writes is wider than the determinant — at every size to sixty-four, at a fiftieth of the modular route's work. The swell the Hermite form suffered belonged to doing elimination with one hand.