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

• Definition:Eigenvector
• Definition:Eigenvalue

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

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $14.1$: The Resolvent and Spectrum