# Definition:Eigenvector

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## Contents

## Definition

Let $H$ be a Hilbert space over $\Bbb F \in \set {\R, \C}$.

Let $A \in \map B H$ be a bounded linear operator.

Let $\alpha \in \Bbb F$ be an eigenvalue of $A$.

A nonzero vector $h \in H$ is said to be an **eigenvector for $\alpha$** if and only if:

- $h \in \map \ker {A - \alpha I}$

That is, if and only if $Ah = \alpha h$.

### Eigenspace

The **eigenspace** for an eigenvalue $\alpha$ is the set $\map \ker {A - \alpha I}$.

By Kernel of Linear Transformation is Closed Linear Subspace, it is a closed linear subspace of $H$.

## Also see

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next) $\text {II}.4.9, 13$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**eigenvalue, eigenvector**