The paper proposes an alternative proof of Lyapunov's matrix inequality about the location of the eigenvalues of a matrix in some region of the complex plane. This new proof does not refer to stability of the trajectories of an associated dynamical system and does not use matrix exponentials. The proposed approach considers the eigenvalue location problem as a mere quadratic optimization problem, which is formulated as an LMI problem with a nonconvex rank constraint. The Lyapunov matrix is regarded as a Lagrange multiplier matrix arising in duality theory.