  We compute the fidelity between the ground states of general quadratic
fermionic hamiltonians and analyze its connections with quantum phase
transitions. Each of these systems is characterized by a $L\times L$ real
matrix whose polar decomposition, into a non-negative $\Lambda$ and a unitary
$T$, contains all the relevant ground state (GS) information. The boundaries
between different regions in the GS phase diagram are given by the points of,
possibly asymptotic, singularity of $\Lambda$. This latter in turn implies a
critical drop of the fidelity function. We present general results as well as
their exemplification by a model of fermions on a totally connected graph.
