  Matrix product states, a key ingredient of numerical algorithms widely
employed in the simulation of quantum spin chains, provide an intriguing tool
for quantum phase transition engineering. At critical values of the control
parameters on which their constituent matrices depend, singularities in the
expectation values of certain observables can appear, in spite of the
analyticity of the ground state energy. For this class of generalized quantum
phase transitions we test the validity of the recently introduced fidelity
approach, where the overlap modulus of ground states corresponding to slightly
different parameters is considered. We discuss several examples, successfully
identifying all the present transitions. We also study the finite size scaling
of fidelity derivatives, pointing out its relevance in extracting critical
exponents.
