Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Hessenberg form — where it appears

An upper triangular matrix with one extra subdiagonal, reached by an orthogonal similarity. Reducing to it costs a fixed multiple of n³ once, and it is what makes each step of the QR algorithm quadratic rather than cubic.

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

-0.16-0.63-0.493.43-0.025-0.632.3-0.69-0.95-1.70.076-0.49-0.694.3-1.6-1.41.93.4-0.95-1.65.40.0141.63-1.7-1.40.0144.60.055-0.0250.0761.91.60.0553.1A, symmetric→-0.164.600004.65.92.500002.51-1.20000-1.24.90.3800000.385.80.2400000.242H = QᵀAQ, tridiagonal‖A − QHQᵀ‖/‖A‖1.1·10⁻¹⁵below the subdiagonal0worst eigenvalue movement7.1·10⁻¹⁵a similarity, so the spectrum is untouched — and every later step is O(n²) rather than O(n³)one reduction, then every iteration is cheapthe eigenvalues did not move

The form that makes it affordable

One Householder reduction, done once, turns every subsequent iteration of the eigenvalue algorithm from cubic to quadratic cost. It changes no answer at all, which is why it is easy to describe as an optimisation and wrong to.

spectra · The QR algorithm 0.46-1.11.5-0.470.2-0.181.90.690.42-0.160.028-0.0160-1.50.160.880.099-0.1100.680.762.30.930.0480-0.250.110.952.11.500001.52.3after reflector 2the highlighted entries are the bulge — the only thing that is not Hessenbergentries below the subdiagonal3reflectors used2reflectors in a whole step5the shifts are never formed — only their sum and productand both of those are real

Two shifts that are never formed

The double shift is defined as a factorisation of (A − μI)(A − μ̄I), which nobody computes. What is computed is the first column of that product — three numbers — and the bulge those three numbers create, pushed down the subdiagonal by n − 2 reflectors until it falls off the bottom.

spectra · Francis

Named alongside it

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

DeflationHouseholder reflectionJacobi's eigenvalue methodOrthogonalityBulge chasingError accumulationFrancis's double shiftThe implicit Q theoremResidualShiftsSimilarity transformationTridiagonal

All concepts