An explicit expression providing the symmetric stress tensor T(m) conjugate to the Seth’s strain measure E(m) for each integer m 6= 0 is presented. The result is obtained by exploiting an original approach for the solution of a tensor equation in the unknown T(m) expressed as function of the powers of the right stretch tensor U. The proposed approach is based upon the spectral decomposition of U and exploits some peculiar features of the set of fourth-order tensors obtained as linear combination of dyadic and square tensor products of the eigenprojectors of U. On the basis of such properties it is shown that the unknown T(m) can be expressed in the given reference frame as linear combination of six fourth-order tensors scaled through coefficients which are rational functions of the eigenvalues of U.