The most obvious application for eigenvalues is that if you find the eigendecomposition A=QλQ^{-1}, where λ=diag(eigenvalues) you can cheaply obtain the results of A^n = QλQ^{-1} ... QλQ^{-1} ... QλQ^{-1} = Qλ^nQ^{-1}. A less likely use is for inverting A or solving a linear system of equations with it.