Eigenvector - Imported Knowlpedia Content

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

Eigenvectors for a fixed eigenvalue $\lambda$ form the eigenspace of $\lambda$ , which is a vector space inside $V$ . In an inner product space, geometric conditions like orthogonality often organize eigenvectors of important classes of operators.

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 $c$ .