Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Perron frobenius — where it appears

The theorem that a nonnegative matrix with a strongly connected graph has its spectral radius as a simple eigenvalue with a strictly positive eigenvector. The positivity is a statement about real numbers and does not survive a norm-wise approximation.

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

01428425670849810⁻¹⁶10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²iterationchange between iteratestwo rates, one curveα0.85λ₂(P)0.9predicted rate0.77measured0.77iterations108against the solve10⁻¹⁶the upper dashed line is αᵏthe curve is on the other one

A ranking that is an eigenvector

PageRank is the stationary vector of a walk that follows links with probability α and jumps at random otherwise. The iteration and the elimination agree to 4·10⁻¹⁷. What α is set to changes which pages come third, fourth and fifth.

graph · random walk 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.

Collatz wielandtDangling nodePagerankPower iterationRegularisationStationary distributionTeleportationCentralityCertificateEigenvectorIrreducibilityMixing time

All concepts