Eigenvector

Definition

Let $F$ be a field, let $V$ be a vector space over $F$, let $T \colon V \to V$ be an endomorphism, and let $\lambda \in F$. An eigenvector of $T$ for $\lambda$ is a vector $v \in V$ with $v \ne 0$ and $T(v) = \lambda v$. Then $\lambda$ is an eigenvalue of $T$.

Remarks

Each nonzero multiple of an eigenvector is again an eigenvector for the same eigenvalue. For $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$, the vector $(1, 1)^T$ is an eigenvector for $3$, because $A (1, 1)^T = (3, 3)^T$. The zero vector is never an eigenvector, by convention.

Explanations

L1 human

An eigenvector keeps its line under the map: the map only stretches it, shrinks it or flips it. The stretch factor is the eigenvalue. Consider a reflection of the plane in a line. The vectors on the line are eigenvectors for $1$, and the vectors perpendicular to it are eigenvectors for $-1$.