Definition:Eigenvector/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.

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:

$A v = \lambda v$

Eigenspace

We say that $\map \ker {A - \lambda I}$ is the eigenspace corresponding to the eigenvalue $\lambda$.

Also see

• Definition:Eigenvalue 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, eigenvector
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): eigenvalue, eigenvector