Eigenspace

An eigenspace of a linear operator $T:V\to V$ associated to a scalar $\lambda\in\mathbb{F}$ is the set

Equivalently,

where $I$ is the identity operator on $V$.

If $\lambda$ is an eigenvalue , then $E_\lambda(T)$ contains nonzero eigenvectors ; otherwise it is $\{0\}$. In all cases, $E_\lambda(T)$ is a vector space under the operations inherited from $V$.

Examples:

• For the identity operator $I$ on $V$, the eigenspace for $\lambda=1$ is all of $V$.
• For the projection $P(x,y)=(x,0)$ on $\mathbb{R}^2$, the eigenspace for $\lambda=1$ is $\{(x,0):x\in\mathbb{R}\}$ and for $\lambda=0$ is $\{(0,y):y\in\mathbb{R}\}$.
• For a diagonal matrix $\operatorname{diag}(d_1,\dots,d_n)$, the eigenspace for $\lambda$ is spanned by those standard basis vectors whose diagonal entry equals $\lambda$.