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Concept

Invariant direction

A line through the origin that a linear map sends into itself, stretched but never turned. It is an eigenvector by another name, and the factor by which the map stretches it is the eigenvalue.

Named by 3 essays across one field — 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 unit square, mapped: area × 5. The unit square and the parallelogram it becomes under a linear map, with the area of that parallelogram computed from its own corners and set against ad − bc.

The number that says how much room is left

A linear map takes the unit square to a parallelogram. The area of that parallelogram is one number, it is computable from the four entries of the matrix, and almost everything the determinant is used for is a restatement of that sentence.

algebra · determinant 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.

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.

algebra · eigenvectors

Named alongside it

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

DeterminantEigenvalueMatrixShearBasisCharacteristic polynomialEigenvectorAreaDefective matrixDiagonalisationOrientationOrthogonality

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