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Concept

Diagonalisation

Rewriting a linear map against a basis of its own eigenvectors, so that it becomes a stretch along each axis and mixes nothing. It makes powers of the map cost two multiplications rather than many, and is available exactly when the eigenvectors span the space.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

The directions the map leaves alone. Unit vectors and their images under the map. On the two marked lines the image points the same way as the original, stretched by 3.00 and 1.00.

The directions a map leaves alone

Almost every arrow comes out of a transformation pointing somewhere else. A few come out pointing exactly where they went in, only longer or shorter. Those few decide nearly everything the map does.

algebra · Eigenvectors The same map, written in the basis of its own eigenvectors. Three panels: the map [2, 1, 1, 2] on the standard grid, the diagonal stretch by 3 and 1 it becomes on the eigenvector grid, and the two put back together.

The same map in a better basis

Measured along its own invariant directions, a linear map stops shearing and becomes two independent stretches. Nothing about the map has changed; the grid it is described against has.

algebra · Eigenvectors The level curve, and the axes the matrix chooses. The curve xᵀAx = 1 for the matrix [2, 0.8, 0.8, 1.4], drawn by solving for the radius at each angle, with the two eigen-directions marked; they cross at a right angle and are the axes of the curve.

Symmetry forces a right angle

A matrix equal to its own reflection across the diagonal always has real stretches and always has perpendicular directions to stretch along. Neither is true of matrices in general, and both follow from one line of algebra.

algebra · Eigenvectors The fast-growing hierarchy at its first few ordinals. A table of the fast-growing hierarchy: one row per ordinal index, one column per argument, with the cells too large to evaluate marked as such.

An ordinal as a growth rate

Index a family of functions by the ordinals, each one iterating the last, and the index becomes a measure of how fast a function grows. The point where the index leaves what arithmetic can prove is exactly where the Goodstein sequence became unprovable.

logic · Ordinals Divergent and convergent integrands pressing towards each other. x·f(x) on a log scale for 1/x, 1/(x ln x), 1/(x ln x ln ln x) (divergent) and 1/(x (ln x)²), 1/(x ln x (ln ln x)²) (convergent), for x up to 10^300.

No integrand sits on the border

The area under 1/x out to infinity is infinite and the area under 1/x² is finite, so it is natural to look for the dividing line. There is none. Divide any divergent integrand by its own running integral and it still diverges, more slowly; divide any convergent one by the square root of its tail and it still converges, more slowly. Paul du Bois-Reymond proved in 1873 that no single integrand can separate the two families — and the slow ones are slow beyond imagining.

analysis · The integral

Named alongside it

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

BasisEigenvalueEigenvectorIterationMatrixOrthogonalityChange of basisCharacteristic polynomialConic sectionConvergenceConvergence rateDefective matrix

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