Would it be correct to say that the eigenspace corresponding to an eigenvalue $\lambda$ for a matrix $A$ is simply the null space of the matrix $A-\lambda I$? My justification is that the eigenspace is made up of every vector $x$ such that $Ax - \lambda I=0$, but if $x$ satsifies that equation then it is in the null space of $A-\lambda I$.
Is there a difference between the two that I'm missing? Any help is appreciated.
By definition, the eigenspace is the space of all vectors $v$ satisfying $Av=\lambda v$. This is equivalent to $Av-\lambda v=0$ or $(A-\lambda I)v=0$. So the eigenspace of $\lambda$ is equal to the nullspace of $A-\lambda I$, as you stated.