Definition:Eigenvalue

Let $$A$$ be an $$n \times n$$ matrix and let $$\mathbf{v}$$ be a vector, $$ \mathbf{v} \in \mathbb{R}^n, \mathbf{v}\neq\mathbf{0} $$.

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

The eigenvalues are usually found by solving the characteristic equation of $$A$$, which is given by $$\det(A-\lambda I)=0 \,\!$$