Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Invariant factors — where it appears

The diagonal entries of an integer matrix's Smith normal form, each dividing the next. Their product is the determinant, the number of nonzero ones is the rank over the rationals, and the number not divisible by a prime p is the rank over the field with p elements — so they decide which primes a modular computation may not use.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as smith normal form — the same set of essays touches all of them, so they are one junction rather than several.

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

Named alongside it

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

Exact arithmeticSmith normal formUnimodularDeterminantHermite normal formLatticeModular arithmeticRank is a decisionUnlucky prime

All concepts