Eigenvectors and eigenvalues

Illustration of a transformation (of Mona Lisa): The image is changed in such a way that the red arrow (vector) does not change its direction, but the blue one does. The red vector therefore is an eigenvector of this transformation, the blue one is not. Since the red vector does not change its length, its eigenvalue is 1. The transformation used is called shear mapping.

Linear algebra talks about functions, which are often called transformations. In that context, an eigenvector is a vector, different from the null vector which does not change direction in the transformation (except if it turns the vector exactly around). The vector may change its length, or become null. The value of the change in length of the vector is known as eigenvalue.

Basics

If there exists a matrix A, a scalar λ, and a vector v, then λ is eigenvalue and v is the eigenvector if the following equation is satisfied:

A·v=λ·v

In other words, if matrix A times the vector v is equal to the scalar λ times the vector v, than λ is the eigenvalue of v, where v is the eigenvector.