**How can I better understand eigenvectors?**

When a matrix M acts upon a vector \vec{v} , the result is in general a completely different vector \vec{v}' with a different magnitude pointing in a totally different direction.

M \vec{v} = \vec{v}\,' where \vec{v} \neq \vec{v}\,'

An eigenvector \vec{v}_e is defined to be a vector that does not change direction when acted on by a matrix; however, an eigenvector can change in magnitude. The scale factor by which the magnitude of the eigenvector changes is called the eigenvalue (\lambda).

M \vec{v}_e = \lambda \vec{v}_e