Eigenvector

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

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

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 $c$.