Eigenspace

Let $T$ be a linear operator $T:V\to V$ over a field $F$, and let $\lambda\in F$. The $\lambda$-eigenspace of $T$ is the subset

This is a subspace of $V$: it contains $0$, and it is closed under addition and scalar multiplication (because $T$ is linear).

The scalar $\lambda$ is an eigenvalue of $T$ if and only if $E_\lambda(T)$ contains a nonzero vector. Equivalently,

where $I$ is the identity map on $V$ and $\ker(S)=\{v:S(v)=0\}$ denotes the kernel (null space) of a linear map $S$.

Examples:

• If $T=\lambda I$ on $V$, then $E_\lambda(T)=V$ and $E_\mu(T)=\{0\}$ for $\mu\neq\lambda$.
• For $A=\begin{pmatrix}1&1\\0&1\end{pmatrix}$ on $F^2$, $E_1(A)=\{(x,0):x\in F\}$.
• For the rotation of $\mathbb{R}^2$ by $90^\circ$, $E_\lambda(T)=\{0\}$ for every real $\lambda$ (no real eigenvalues).