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We propose that there is a unique expansion for the string theory S-matrixelements of tachyons that corresponds to non-abelian tachyon action. For thoseS-matrix elements which, in their expansion, there are the Feynman amplitudesresulting from the non-abelian kinetic term, we give a prescription on how tofind the expansion. The gauge invariant action is an $\alpha'$ expanded action,and the tachyon mass $m$ which appears as coefficient of many differentcouplings, is arbitrary. We then analyze in details the S-matrix element offour tachyons and the S-matrix element of two tachyons and two gauge fields, inboth bosonic and superstring theories, in favor of this proposal. In thesuperstring theory, the leading terms of the non-abelian gauge invariantcouplings are in agreement with the symmetrised trace of the direct non-abeliangeneralization of the tachyonic Born-Infeld action in which the tachyonpotential is consistent with $V(T)=e^{\pi\alpha' m^2T^2}$. In the bosonictheory, on the other hand, the leading terms are those appear in superstringcase as well as some other gauge invariant couplings which spoils thesymmetrised trace prescription. These latter terms are zero in the abeliancase.Comment: Latex, 27 pages, no figures,v4:change the introduction section, add  some notes to clarify the idea, add reference

S-matrix elements and off-shell tachyon action with non-abelian gauge symmetry