eigenvalue

$\lambda$ in $A\mathbf v = \lambda\mathbf v$ — the stretch factor belonging to an eigenvector.

The scalar $\lambda$ in $A\mathbf v=\lambda\mathbf v$ is the eigenvalue belonging to eigenvector $\mathbf v$ : $\lambda>1$ stretches $\mathbf v$ , $0<\lambda<1$ shrinks it, $\lambda<0$ flips it to the opposite ray, and $\lambda=0$ collapses $\mathbf v$ to the origin (meaning $\mathbf v$ lies in $A$ 's null space). $\lambda$ itself carries no restriction to being nonzero.