Eigenspace

Definition

Let $F$ be a field, let $V$ be a vector space over $F$, let $T \colon V \to V$ be an endomorphism, and let $\lambda \in F$. The eigenspace of $T$ for $\lambda$ is $$E_\lambda = \ker(T - \lambda \id_V) = \{v \in V : T(v) = \lambda v\}.$$

Remarks

The eigenspace is a subspace of $V$; it is the kernel of a linear map. Its nonzero elements are the eigenvectors of $T$ for $\lambda$. $E_\lambda \ne 0$ if and only if $\lambda$ is an eigenvalue of $T$. The dimension of $E_\lambda$ is the geometric multiplicity of $\lambda$.

Explanations

L1 human

The eigenspace collects all the vectors that the map stretches by the same factor $\lambda$, with the zero vector. These vectors form a subspace, for example a line or a plane. On this subspace, the map acts as multiplication by $\lambda$.