What is an eigenvalue?

If $A$ is any $n \times n$ square matrix, then the eigenvalues of A are those values $\lambda$ for which the determinant $\det \left(A - \lambda I\right) = 0$, (the zero matrix), where $I$ is the $n \times n$ identity matrix.
The corresponding vectors $x$ such that $A x = \lambda x$ are called the eigenvectors corresponding to the eigenvalues $\lambda$.