Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Unimodular — where it appears

An integer matrix whose determinant is plus or minus one, and therefore whose inverse is also integral. It is the class of transformation that changes a description without changing the lattice it describes, which is what makes the lattice determinant a conserved quantity that an implementation can be checked against.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

ℚ6𝔽24𝔽35𝔽56𝔽76𝔽116𝔽136𝔽1016𝔽655376rank, by the ring the entries are read inthe rank of one matrixover ℚ6over 𝔽24over 𝔽35over 𝔽56over 𝔽76nothing is rounded hereand the answer still is not a property of the matrix

The rank depends on the ring

A floating-point rank is a decision about a threshold. Remove the arithmetic error entirely and the threshold goes away — and the answer still is not a property of the array of numbers, because one integer matrix has rank six over the rationals, five modulo three and four modulo two, with nothing rounded and nothing decided.

exact · Exact rank s10 bitss20 bitss30 bitss40 bitss514 bitsinvariant factors, in bitstwo routes, and a conserved quantitydet A-2.3·10⁴Π invariants2.3·10⁴SNF widest15HNF widest24det of the transform-1an algorithm and a definitionagreeing as integers

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.

exact · Normal forms the same lattice, twicewhat the reduction may not changedet, before1det, after1defect, before7.1defect, after1LLL steps1the determinant is the invariantand the defect is what is being reduced

A basis that describes its lattice badly

The same set of points has infinitely many bases, they are all correct, and they are not equally useful. One measurement separates them — the product of the vectors' lengths over the lattice determinant — and the determinant is the invariant the reduction may not change, which is what makes the reduction checkable.

exact · Lattice reduction

Named alongside it

The objects these essays reach for when they reach for this one.

Exact arithmeticDeterminantInvariant factorsLatticeSmith normal formHermite normal formLattice reductionModular arithmeticOrthogonality defectRank is a decisionRational reconstructionUnlucky prime

All concepts