# Definition:Eigenvector

## Definition

### Eigenvector of Linear Operator

Let $K$ be a field.

Let $V$ be a vector space over $K$.

Let $A : V \to V$ be a linear operator.

Let $\lambda \in K$ be an eigenvalue of $A$.

A non-zero vector $v \in V$ is an eigenvector corresponding to $\lambda$ if and only if:

$v \in \map \ker {A - \lambda I}$

where:

$I : V \to V$ is the identity mapping on $V$
$\map \ker {A - \lambda I}$ denotes the kernel of $A - \lambda I$.

That is, if and only if:

$A v = \lambda v$

### Eigenvector of Real Square Matrix

Let $\mathbf A$ be a square matrix of order $n$ over $\R$.

Let $\lambda \in \R$ be an eigenvalue of $\mathbf A$.

A non-zero vector $\mathbf v \in \R^n$ is an eigenvector corresponding to $\lambda$ if and only if:

$\mathbf A \mathbf v = \lambda \mathbf v$

## Also see

• Results about eigenvectors can be found here.