Concept
Characteristic polynomial
The polynomial whose roots are the eigenvalues of a square matrix, got as the determinant of that matrix with a variable taken off the diagonal. Its coefficients are the trace, the determinant and the quantities between them, so it records only what a change of basis leaves alone.
Named by 2 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 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.
Named alongside it
The objects these essays reach for when they reach for this one.
DeterminantEigenvalueEigenvectorInvariant directionMatrixShearBasisDefective matrixDiagonalisationOrthogonalityPolynomialTrace