Eigenspace - Imported Knowlpedia Content

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

E_\lambda(T)=\{v\in V:\ T(v)=\lambda v\}.

Equivalent characterizations

Equivalently,

E_\lambda(T)=\{v\in V:\ (T-\lambda I)(v)=0\},

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

Remarks

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 $\lambda$ .