In this paper we show how to decompose the singularly perturbed algebraic Riccati equation and the corresponding linear-quadratic optimal control problem at steady state in terms of reduced-order pure-slow and pure-fast problems by using the eigenvector approach. The eigenvector approach should be used for decomposition of singularly perturbed control systems in the cases when the singular perturbation parameter is not very small. In such cases the decomposition methods based on series expansions, fixed point iterations, subspace iterations, and Newton iterations, either fail to produce solutions of the corresponding algebraic equations or display very slow convergence.