Eigenvector

Let $T$ be a linear operator $T:V\to V$ over a field $F$. A nonzero vector $v\in V$ is an eigenvector of $T$ if there exists a scalar $\lambda\in F$ such that

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

For a fixed $\lambda$, the collection of all eigenvectors with eigenvalue $\lambda$, together with the zero vector, forms the eigenspace for $\lambda$.

Examples:

• For $T=\lambda I$ on $F^n$, every nonzero vector is an eigenvector with eigenvalue $\lambda$.
• For $T(x,y)=(x,0)$ on $\mathbb{R}^2$, the vectors $(1,0)$ and $(2,0)$ are eigenvectors with eigenvalue $1$, while $(0,1)$ is an eigenvector with eigenvalue $0$.
• For $A=\begin{pmatrix}1&1\\0&1\end{pmatrix}$ on $F^2$, the vector $(1,0)$ is an eigenvector with eigenvalue $1$, but $(0,1)$ is not an eigenvector.