Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Collatz wielandt — where it appears

The characterisation bracketing the Perron root between the smallest and largest of the ratios (Ax)_i over x_i for any strictly positive x. It is a two-sided bound that is proved rather than estimated, from one matrix-vector product.

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

02244668811013215417610⁻¹⁶10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²iterationchange between iteratestwo rates, one curveα0.85λ₂(P)1predicted rate0.85measured0.84iterations176against the solve1.4·10⁻¹⁶the upper dashed line is αᵏthe curve is on the other one

A chain with no stationary vector

A page with no outgoing links loses forty per cent of the walker's probability in six hundred steps. A directed cycle never converges at all. And on a graph whose links only run one way, the entire rank of half the vertices is exactly one minus the teleportation parameter.

graph · random walk 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

Named alongside it

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

Perron frobeniusCentralityCertificateDangling nodeEigenvectorIrreducibilityMixing timePagerankPeriodicityPower iterationRegularisationSpectral radius

All concepts