The eigenvalue complementarity problem (EiCP) is a kind of very useful model, which is widely used in the study of many problems in mechanics, engineering, and economics. The EiCP was shown to be equivalent to a special nonlinear complementarity problem or a mathematical programming problem with complementarity constraints. The existing methods for solving the EiCP are all nonsmooth methods, including nonsmooth or semismooth Newton type methods. In this paper, we reformulate the EiCP as a system of continuously differentiable equations and give the Levenberg-Marquardt method to solve them. Under mild assumptions, the method is proved globally convergent. Finally, some numerical results and the extensions of the method are also given. The numerical experiments highlight the efficiency of the method.