In this paper, we derived a new piezoresistance tensor equation for a cubic single crystal. This equation can be expressed in terms of the three independent principal components of piezoresistance tensors, the isotropic and deviatoric stress tensors, and the fourth-rank coordinate transformation tensor. The piezoresistance tensor equation can be decomposed into a relation between only the hydrostatic part of the stress tensor and the trace of the resistivity change tensor and into a relation between only the deviatoric parts. The hydrostatic part of the piezoresistance tensor equation is invariant with respect to a coordinate transformation. On the other hand, the deviatoric part of the tensor equation is traceless. The proposed piezoresistance tensor decomposition gives a new physical insight into the classical theory of Pfann and Thurston [J. Appl. Phys. 32 (2008) 1961]. It was shown that Pfann and Thurston's theory can be rewritten as a special case of our tensor decomposition. Furthermore, to demonstrate the consistency between the proposed tensor equation and the experimental evidence, some basic experiments on a single-crystal silicon piezoresistive rosette stress gauge subjected to multiaxial stress were carried out.