We present an iterative method based on repeatedly inverting the Monge–Ampère operator with Dirichlet boundary condition and prescribed right-hand side on a bounded, convex domain Ω ⊂ R n \Omega \subset \mathbb {R}^{n} . We prove that the iterates u k u_k generated by this method converge as k → ∞ k \to \infty to a solution of the Monge–Ampère eigenvalue problem { d e t D 2 u = λ M A ( − u ) n a m p ; in  Ω , u = 0 a m p ; on  ∂ Ω . \begin{equation*} \begin {cases} \mathrm {det} D^2u = \lambda _{MA} (-u)^n & \quad \text {in } \Omega ,\\ u = 0 & \quad \text {on } \partial \Omega . \end{cases} \end{equation*} Since the solutions of this problem are unique up to a positive multiplicative constant, the normalized iterates u ^ k ≔ u k | | u k | | L ∞ ( Ω ) \hat {u}_k \coloneq \frac {u_k}{||u_k||_{L^{\infty }(\Omega )}} converge to the eigenfunction of unit height. In addition, we show that lim k → ∞ R ( u k ) = lim k → ∞ R ( u ^ k ) = λ M A \lim _{k \to \infty } R(u_k) = \lim _{k \to \infty } R(\hat {u}_k) = \lambda _{MA} , where the Rayleigh quotient R ( u ) R(u) is defined as R ( u ) ≔ ∫ Ω ( − u )   d e t D 2 u ∫ Ω ( − u ) n + 1 . \begin{equation*} R(u) \coloneq \frac {\int _{\Omega } (-u) \ \mathrm {det}D^2u}{\int _{\Omega } (-u)^{n+1}}. \end{equation*} Our method converges for a wide class of initial choices u 0 u_0 that can be constructed explicitly, and does not rely on prior knowledge of the Monge–Ampère eigenvalue λ M A \lambda _{MA} .