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 $A$ :


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

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 $A$ the kernel of $A$ is non-trivial.

Also see

 * Definition:Eigenvector of Linear Operator