  Synchronization is studied in an array of identical oscillators undergoing
small vibrations. The overall coupling is described by a pair of
matrix-weighted Laplacian matrices; one representing the dissipative, the other
the restorative connectors. A construction is proposed to combine these two
real matrices in a single complex matrix. It is shown that whether the
oscillators synchronize in the steady state or not depends on the number of
eigenvalues of this complex matrix on the imaginary axis. Certain refinements
of this condition for the special cases, where the restorative coupling is
either weak or absent, are also presented.
