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Let Mn(C) denote the algebra of all n × n complex matrices, and fix a nonzero vector x0 in Cn. For any matrix T ∈Mn(C), let σT (x0) denote the local spectrum of T at x0. Given three scalars μ, ν and ξ simultaneously nonzero, we study maps ϕ on Mn(C) satisfying σμSTS+νTS+ξST (x0) = σμϕ(S)ϕ(T)ϕ(S)+νϕ(T)ϕ(S)+ξϕ(S)ϕ(T)(x0) for all S, T ∈Mn(C). Our main result extends and unifies the main results of several papers on maps on Mn(C) preserving the local spectrum of different products. [ABSTRACT FROM AUTHOR]