Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Spectral radius — where it appears

The largest eigenvalue modulus, which decides whether the powers of a matrix decay in the end and nothing about what they do first. It answers one question — do the powers decay eventually — and it is routinely read as answering a different one about what they do meanwhile.

Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.

015030045060075090010⁻¹³10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹iteration‖r‖ / ‖b‖JacobiGauss–SeidelSOR ω=1.78closed form vs measuredρ Jacobi, exact0.99ρ measured0.99ρ Gauss–Seidel, exact0.981D Laplacian, n = 24ω optimal = 1.777

A rate that is known in advance

On the model problem, Jacobi contracts by cos(π/(n+1)) per step, Gauss–Seidel by its square, and optimally relaxed SOR by a number given in closed form. Three rates, all known before anything runs, and all measurable against what runs.

iterative · Stationary 1611162126313610⁻²²10⁻¹⁹10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹vertex, clique first then tail|entry| relative to the largestone rounding of the largest entrya bound that is provedPerron root11bracket, low11bracket, high11bracket width1.2·10⁻⁴smallest entry-1.5·10⁻²⁰entries below zero4every entry is positiveand the picture disagrees

An eigenvector that must not change sign

Perron's theorem says the leading eigenvector of a connected nonnegative matrix is strictly positive. On a clique with a long tail, four of its thirty-six entries come back negative — and beside them is the one two-sided bound on this site that is proved rather than estimated.

graph · Perron frobenius 027548110813510⁻⁶10⁻⁴10⁻²110²10⁴10⁶power‖Aᵏ‖Kreiss constant 6760e · n · K‖Aᵏ‖ρᵏtwo routes to one peakspectral radius0.8peak of ‖Aᵏ‖2·10⁴Kreiss constant6757e · n · K1.1·10⁵everything here decays in the endand one of these curves says how much first

A spectral radius that grows first

ρ(A) below one guarantees that the powers of A go to zero and says nothing about what they do on the way. Here they rise by a factor of twenty thousand before turning over, and the peak is bracketed above and below by a constant computed from the resolvent norms outside the unit circle — two routes to one number, one through the plane and one through the powers.

spectra · Non-normality

Named alongside it

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

Asymptotic analysisCentralityCertificateCollatz wielandtConjugate gradientsConvergence rateEigenvectorGauss–SeidelJacobi iterationThe Kreiss constantKrylov subspaceModel problem

All concepts