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 $A$.

Also see

• Definition:Eigenvector of Linear Operator

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text {II}.4.9$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): eigenvalue
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): eigenvalue
• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $14.1$: The Resolvent and Spectrum