  We derive a general method for determining the necessary and sufficient
conditions for exact factorization $|\Psi\rangle=\otimes_p |\psi_p\rangle$ of
an eigenstate of a many-body Hamiltonian $H$, based on the quantum covariance
matrix of the relevant local operators building the Hamiltonian. The "site" $p$
can be either a single component or a group of subsystems. The formalism is
then used to derive exact dimerization and clusterization conditions in spin
systems, covering from spin-$s$ singlets and clusters coupled to $0$ total spin
to general nonmaximally entangled spin-$s$ dimers (generalized singlets). New
results for field induced dimerization in anisotropic $XYZ$ arrays under a
magnetic field are obtained.

Quantum Physics