Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Eigenvectors — where it appears

The directions a matrix only stretches, whose sensitivity is governed by the gap to the next eigenvalue rather than by the eigenvalue's own conditioning.

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

036912151821242700.250.50.7511.25index kweight appliedonebidiagonalArnoldiis it a function of σmisfit, even fit1.6·10⁻¹²misfit, general fit0.063‖A − Aᵀ‖/‖A‖0.086one method's weights do not notice the operatorand the other's stop being a function of σ

The basis decides what a filter is

The vocabulary of regularisation is spectral — a method keeps a component or discards it, and the weights are a function of the singular value. Row-normalising a symmetric blur so that it preserves a constant makes it 8.6% asymmetric, and that is enough to move GMRES's weights from 7·10⁻¹⁴ off a function of σ to 4.4·10⁻².

regularisation · gmres 10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²10⁻¹⁴10⁻¹10¹²10²⁵10³⁸10⁵¹10⁶⁴δ, the gap between consecutive eigenvaluesrelative error in e^Aan answer with no correct digitsV f(Λ) V⁻¹scaling and squaring‖A_δ − A₀‖exact eigenvalues throughoutκ(V) at the smallest δ3.3·10⁸²eigen route2.9·10⁶⁵scaling and squaring4.1·10⁻¹²distance to the limit7·10⁻¹²the eigenvalues are the diagonaland they are exact at every stop

A function of a matrix is not a function of its entries

Everybody learns that f(A) means diagonalise, apply f to the eigenvalues, undiagonalise. That is a definition, not a method. On a matrix seven picometres from a defective one — with exact eigenvalues and eigenvectors from a closed form — the definition returns an answer wrong by sixty-five orders of magnitude, and a method that never mentions an eigenvalue returns the right one.

spectra · matrix function

Named alongside it

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

Arnoldi iterationCondition numberDefective matrixEigenvaluesExact ground truthFilter factorsGMRESIll posed problemJordan formKrylov subspaceLsqrMatrix exponential

All concepts