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In this paper a mixed complementarity eigenproblem (MCEIP) is formulated and a method is proposed for its numerical solution. This mathematical problem is motivated by the study of divergence instabilities of static equilibrium states of finite dimensional mechanical systems with unilateral frictional contact. The complementarity eigenproblem is solved by transforming it into a non-monotone mixed complementarity problem (MCP), which is then solved by using the algorithm PATH. The proposed method is used to study some small sized examples and some large finite element problems.