Eigenvalues and Eigenvectors

Eigenvalues together with eigenvectors define aspects of linear transforms on spaces:

An eigenvector defines a direction in which a space is scaled by a transform.
An eigenvalue defines a length of scaled change related to the eigenvector.

Shear Mapping Example

Below, in a shear mapping linear transform.

The red arrow changes direction but the blue arrow does not.

The blue arrow is an eigenvector of this shear mapping because it doesn't change direction, and since its length is unchanged, its eigenvalue is 1.