Concept

Eigenvalue

The factor by which a linear map stretches one of the directions it does not turn.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as invariant direction — the same set of essays touches all of them, so they are one junction rather than several.

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 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

Named alongside it

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

Basis Determinant Invariant direction Matrix Shear Area Characteristic polynomial Diagonalisation Eigenvector Orientation Orthogonality Scaling

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