An eigenvector of an $n \times n$ matrix $A$ is a vector $\vec v \in \R^n$ such that
Basically, an eigenvector is a vector that only gets scaled by $A$, excluding the trivial case where the vector is $\vec 0$. And, the scaling factor $\lambda$ is called an eigenvalue of $A$.
A diagonal matrix is an $n \times n$ matrix that has zeroes that is not the main diagonal.
Fact: For a diagonal $n \times n$ matrix $A$, the standard basis vectors $\{ \vec e_1, \cdots, \vec e_n \}$ are all the eigenvectors of $A$ with associated eigenvalues being the diagonal entries (for all non-zero entries).
$$ \text{ $\lambda$-eigenspace of A $:=$ $\{$eigenvectors of $A$ with eigenvalues } \lambda \} \cup \{ \vec 0 \} \subset \R $$
Notice that the $\lambda$-eigenspace of $A$ includes the trivial $\vec 0$.
Fact:
$$ \text{$\lambda$-eigenspace of }A = \text{Nul}(A - \lambda I_n) $$
If there exists an eigenvalue $\lambda$ of $A$, then we can say the null-space of $A - \lambda I_n$ is non-zero
$$ \exists \lambda \text{ eigenvalue of } A \iff \text{Nul} (A - \lambda I_n) \neq \{ \vec 0 \} $$