# Definition:Eigenvalue

## General 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.

A scalar $\alpha \in \Bbb F$ is said to be an **eigenvalue** of $A$ if and only if:

- $\map \ker {A - \alpha I} \ne \paren {\mathbf 0_H}$

That is, if and only if the bounded linear operator $A - \alpha I$ has nontrivial kernel.

### Point Spectrum

The set of all **eigenvalues** of $A$ is denoted $\map {\sigma_p} A$.

In the field of functional analysis, it is frequently called the **point spectrum of $A$**.

## Definition in $\R^n$

Let $\mathbf A$ be an square matrix of order $n$, and let $\mathbf v$ be a vector, $ \mathbf v \in \R^n, \mathbf v \ne \mathbf 0$.

If $\mathbf A \mathbf v = \lambda \mathbf v$ for some $\lambda\in \R$, which is a scalar, then $\lambda$ is called an **eigenvalue** of $\mathbf A$ with a corresponding eigenvector $\mathbf v$.

The **eigenvalues** are usually found by solving the characteristic equation of $\mathbf A$, which is given by:

- $\map \det {\mathbf A - \lambda \mathbf I} = 0$

## Also see

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next) $II.4.9$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**eigenvalue** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**eigenvalue** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**eigenvalue, eigenvector**