Definition:Eigenvalue

General Definition
Let $H$ be a Hilbert space over $\Bbb F \in \left\{{\R, \C}\right\}$.

Let $A \in B \left({H}\right)$ be a bounded linear operator.

A scalar $\alpha \in \Bbb F$ is said to be an eigenvalue of $A$ iff $\operatorname{ker} \left({A - \alpha I}\right) \ne \left({\mathbf{0}_H}\right)$.

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

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:
 * $\det \left({\mathbf A - \lambda \mathbf I}\right) = 0$