The authors consider the Monge-Ampère eigenvalue problem: Find a convex function \(u \in C^2(\Omega) \cap C(\overline{\Omega})\) and a positive number \(\lambda_{MA}\) such that \(\mbox{det}D^2u = \lambda_{MA} (-u)^n\) in \(\Omega\) and \(u = 0\) on \(\partial \Omega\) hold, where \(\Omega\) is a bounded, convex domain in \(\mathbb{R}^n\). Some results on the existence of a solution of this problem and two methods for the solution of this problem, which are known from the literature, are summarized (see, [P.-L. Lions, Ann. Mat. Pura Appl. (4) 142, 263–275 (1985; Zbl 0594.35023)], [K. Tso, Invent. Math. 101, No. 2, 425–448 (1990; Zbl 0724.35040)], and [N. Q. Le, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 18, No. 4, 1519–1559 (2018; Zbl 1478.47011)]). The authors propose the iterative method \(\mbox{det}D^2 u_{k+1} = R(u_k)(-u_k)^n\) in \(\Omega\), \(u_{k+1} = 0\) on \(\partial \Omega\) for constructing a sequence of functions \(u_k\). Hereby, \(u_0\) is an initial guess satisfying some conditions and \(R(u_k)\) denotes the Rayleigh quotient \(\int_\Omega (-u_k) \mbox{det}D^2 u_k/ \int_\Omega (-u_k)^{n+1}\). It is proved that the sequence \(\{u_k\}\) converges uniformly to a nonzero Monge-Ampère eigenfunction and that \(\lim_{k\rightarrow \infty} R(u_k) = \lambda_{MA}\).