Eigenvector - Knowlpedia

An eigenvector of a linear operator $T:V\to V$ is a nonzero vector $v\in V$ for which there exists a scalar $\lambda\in\mathbb{F}$ such that

T(v)=\lambda v.

The corresponding scalar $\lambda$ is an eigenvalue of $T$ .

Remarks

The eigenspace for an eigenvalue $\lambda$ is $\ker(T-\lambda I)$ . Its nonzero vectors are precisely the eigenvectors with eigenvalue $\lambda$ ; the zero vector belongs to the eigenspace but is not an eigenvector.

Examples

For $A=\operatorname{diag}(2,3)$ on $\mathbb{R}^2$ , the vector $(1,0)$ is an eigenvector with eigenvalue $2$ .
For the projection $P(x,y)=(x,0)$ , the vector $(1,0)$ is an eigenvector with eigenvalue $1$ and $(0,1)$ is an eigenvector with eigenvalue $0$ .
For the scaling map $T(v)=c\,v$ , every nonzero vector is an eigenvector with eigenvalue $c$ .