Definition

For $T\in\mathscr{L}(V)$, and eigenvector of $T$ is a non-zero vector $v\in V$ such that $T(v)=\lambda v$ for some $\lambda\in\mathbb{F}$. The value $\lambda$ is called an eigenvalue of $T$. If $\lambda$ is an eigenvalue of $T$, then the set of eigenvectors $$W_\lambda=\{v\in V|T(v)=\lambda v\}$$is a subspace of $V$, and we call $W_\lambda$ the eigenspace corresponding to the eigenvalue $\lambda$.

Remark

We can interpret the subspace $W_\lambda$ as the kernel of the map $T-\lambda I_V$. If $(T-\lambda I_V)(v)=0$, then we can rearrange this equation to get $T(v)=\lambda I_V(v)=\lambda v$. Therefore $\mbox{Ker}(T-\lambda I_V)=W_\lambda$.