Concept
Defective matrix
A square matrix with fewer independent eigenvectors than its size, so that no change of basis makes it diagonal. The shortfall is always at a repeated eigenvalue, and the nearest available form keeps ones just above the diagonal to record what is missing.
Named by 2 essays across one field — each of them below, with the objects they name alongside it.
![The polynomial whose roots are the stretches. The determinant of A − λI plotted against λ for the map [2, 1, 1, 2], with its roots at 3 and 1 marked.](/f/the-polynomial-whose-roots-are-the-stretches-1b3407e2.svg)
The polynomial whose roots are the stretches
Finding the directions a map leaves alone means finding the numbers at which it crushes something to nothing. Those numbers are the roots of one quadratic, and everything the map does is written in its two coefficients.
![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.
Named alongside it
The objects these essays reach for when they reach for this one.
EigenvalueEigenvectorMatrixBasisChange of basisCharacteristic polynomialConvergence rateDeterminantDiagonalisationInvariant directionIterationPolynomial