eigenspace

$\operatorname{Nul}(A-\lambda I)$ — every eigenvector for a given $\lambda$ , plus $\mathbf 0$ .

For an eigenvalue $\lambda$ of $A$ , the eigenspace is $\operatorname{Nul}(A-\lambda I_n)$ : every vector, including $\mathbf 0$ , solving $(A-\lambda I)\mathbf v=\mathbf 0$ . Being a null space, it's automatically a subspace. Its dimension is the eigenvalue's geometric multiplicity.