# Definition:Eigenvalue/Linear Operator

## Definition

Let $K$ be a field.

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

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

$\lambda \in K$ is an eigenvalue of $A$ if and only if:

$\map \ker {A - \lambda I} \ne \set {0_V}$

where:

$0_V$ is the zero vector of $V$
$I : V \to V$ is the identity mapping on $V$
$\map \ker {A - \lambda I}$ denotes the kernel of $A - \lambda I$.

That is, $\lambda \in K$ is an eigenvalue of $A$ if and only if the kernel of $A - \lambda I$ is non-trivial.

### Point Spectrum

Let $\map {\sigma_p} A$ be the set of eigenvalues of $A$.

We call $\map {\sigma_p} A$ the point spectrum of $A$.