Eigenvectors and eigenvalues

An eigenvector $\mathbf{v}$ of a matrix $\mathbf{M}$ is one whose direction is unchanged by $\mathbf{M}$. For the purposes of eigenvectors, a vector being reversed (e.g. $\mathbf{v}$ to $-2\mathbf{v}$) counts as its direction being unchanged.

Stated differently: if $\mathbf{v}$ is an eigenvector of $\mathbf{M}$, then $\mathbf{Mv}=\lambda \mathbf{v}$ for some scalar $\lambda$. Here, $\lambda$ is an eigenvalue of $\mathbf{M}$ - specifically the eigenvalue corresponding to $\mathbf{v}$.