  We prove Pfister's local-global principle for hermitian forms over Azumaya
algebras with involution over semilocal rings, and show in particular that the
Witt group of nonsingular hermitian forms is $2$-primary torsion. Our proof
relies on a hermitian version of Sylvester's law of inertia, which is obtained
from an investigation of the connections between a pairing of hermitian forms
extensively studied by Garrel, signatures of hermitian forms, and positive
semidefinite quadratic forms.
