Definition:Eigenvalue/Linear Operator

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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 $A$.


Also see

  • Results about eigenvalues can be found here.


Sources