Diagonalisation
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
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.
![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.](/f/the-same-map-written-in-the-basis-of-its-own-eigenvectors-20597990.svg)
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.
![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.](/f/the-level-curve-and-the-axes-the-matrix-chooses-db3c3a9d.svg)
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.
Named alongside it
The objects these essays reach for when they reach for this one.
BasisEigenvalueEigenvectorMatrixOrthogonalityChange of basisCharacteristic polynomialConic sectionConvergence rateDefective matrixDeterminantInner product