Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Quadratic form — where it appears

The scalar x^T A x, read as a function of the vector rather than of the matrix. For a graph Laplacian it is a sum over edges of squared differences, so it is a statement about the graph, and it is what a sparsifier and a partition are both defined by.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

the matrix, measuredvertices40edges223‖L·1‖∞0zero eigenvalues1components, by search1λ₂1.5laid out at its own eigenvectorsand the row sums are exactly zero

A matrix with no numbers in it

A graph arrives as vertices and edges. Two different matrices can be built from it, they answer different questions, and one of them has a null vector that is exact — the only object in these essays whose kernel is known before anything runs.

graph · Graph laplacian 10⁻³10⁻²10⁻¹110¹12345678910conductance, and the two bounds on itpathcyclegridbarbelltwo blockshypercubepreferentialstarcompletethe bar is the inequalitythe dot is the graph

Two Laplacians of one graph

The combinatorial Laplacian D − A and the normalised one, which conjugates it by the inverse square roots of the degrees, are built from the same object, are not similar to each other, and answer different questions. On a graph whose degrees are equal they coincide. On one whose degrees span an order of magnitude their second eigenvalues are sixteen times apart.

graph · Graph laplacian 19172533414911.251.51.752indexsample ÷ originalkept, and not keptedges kept344of1225eigenvalue ratio, low0.43high1.7degree ratio, low0.56high1.5diameter, after3filled dots are eigenvaluesopen dots are degrees

A graph with a tenth of the edges

Keeping 344 of 1,225 edges, sampled by effective resistance and reweighted, preserves every eigenvalue of the Laplacian to within a factor of 1.7. It preserves no degree — half of them are wrong by more than a third — and it takes the diameter from one to three.

graph · Effective resistance Re λIm λwhat survives the arrowsvertices18arcs18worst row sum0worst column sum0largest |Im λ|0.98asymmetry1the null vector is still exactand nothing else about the spectrum is real

A Laplacian that is not symmetric

Point the edges and the matrix stops being symmetric. Its row sums are still exactly zero, so the null vector survives; everything built on the quadratic form does not, and the eigenvalues need a plane rather than a line. Asymmetry permits that and does not force it, which the smallest case here demonstrates by being asymmetric and real.

graph · Directed laplacian Cheeger's band, for the circulationcirculation · arcsvertices on the smaller sidewhich of the two the theorem is aboutλ₂ of 𝓛0.066λ₂/20.033√(2λ₂)0.36circulation Φ0.12arc conductance0.04their ratio3.1the inequality holdsabout a quantity nobody counts

A conductance the arcs do not measure

Symmetrising a directed Laplacian with respect to its walk recovers everything the arrows took — a real spectrum, a sweep cut, a Cheeger inequality. What it does not recover is the quantity: the inequality bounds the probability that a step of the walk crosses the cut, which on one graph here is three times the weight of the arcs that do.

graph · Directed laplacian 110¹00.30.60.91.2trust-region radius Δshare of the exact model decreasethe exact subproblem7 radii, n = 60share at the smallest radius0.95share at the largest0.3products, at most8radii stopped by the curvature4a few products against an eigendecompositionand most of the decrease

The certificate that arrives soonest is worth least

The more negative a Hessian's smallest eigenvalue, the sooner conjugate gradients meets a direction of negative curvature — and the less of the exact trust-region decrease that direction turns out to be worth. At λₘᵢₙ = −10 the step arrives after two products and gets 39.6 per cent; at −10⁻³ the same two products get 89.8, and the whole sweep costs eight.

iterative · Trust-region

Named alongside it

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

Graph laplacianAlgebraic connectivityConductanceDegree sequenceDirected graphNull spaceStationary distributionAdjacency matrixBisectionCertificateCheeger inequalityComplex eigenvalues

All concepts