Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Unlucky prime — where it appears

A prime dividing a quantity a modular computation is trying to recover, at which the residue is zero and carries no information. The elimination reports a singular matrix and is correct about the finite field it is working in, nothing in the arithmetic is inaccurate, and one residue cannot distinguish the case from a genuinely singular matrix.

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

det mod p, as a fraction of p3713a prime that divides the answerdet A3·10⁴primes swept25unlucky5rate0.2det, in bits15singular mod p is not singularand one residue cannot tell them apart

A prime that divides the answer

A modular elimination reports a singular matrix and is telling the truth — over the field with p elements the matrix is singular. Over the rationals it is not. Nothing in the residue distinguishes the two cases, no quantity is small enough to be suspicious, and the wrong answer is a correct computation of a different question.

exact · Modular lift 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

Named alongside it

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

Exact arithmeticModular arithmeticRank is a decisionChinese remainder theoremDeterminantHadamard boundInvariant factorsSmith normal formUnimodular

All concepts