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Let M n (C) denote the algebra of all n × n complex matrices, and x 0 a nonzero vector in C n . For two fixed scalars μ and ν in C for which (μ,ν) = (0,0) , we characterize all maps ϕ on M n (C) satisfyingThis provides, in particular, a complete description of all maps on M n (C) preserving the local spectrum of the skew double product " T S * " or the skew triple product " T S * T " of matrices. It also unifies and extends several known results on local spectrum preservers. (2010): Primary 47B49, Secondary 47A10, 47A11. Mathematics subject classification