Definition:Eigenspace

Definition
Let $K$ be a field.

Let $V$ be a vector space over $K$.

Let $A : V \to V$ be a linear operator.

Let $I : V \to V$ be the identity mapping on $V$.

Let $\lambda \in K$ be an eigenvalue of $A$.

Let $\map \ker {A - \lambda I}$ be the kernel of $A - \lambda I$.

We say that $\map \ker {A - \lambda I}$ is the eigenspace corresponding to the eigenvalue $\lambda$.

Also see

 * Kernel of Linear Transformation is Closed Linear Subspace shows that the eigenspace $\map \ker {A - \lambda I}$ is a closed linear subspace of $V$.