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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