Definition:Eigenvalue

Let $$\mathbf{A}$$ be an $n \times n$ matrix 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:
 * $$\det \left({\mathbf{A} - \lambda \mathbf{I}}\right) = 0 \,\!$$