2.2.3—Eigen terminology

An eigenvector v of A is a non-zero vector satisfying Av=λv; λ is its eigenvalue. The vector’s direction is preserved, though it may be stretched, reversed or collapsed.

Eigenvectors are directions, not arbitrary points. The zero vector is excluded because it would satisfy every λ and carry no directional information.

A diagonal matrix [[3,0],[0,−1]] has eigenvectors along the coordinate axes: the x-direction is scaled by 3 and the y-direction is reversed by factor −1.

An eigenvector is not unique in length, and a repeated eigenvalue can have one or many independent eigendirections.