Abstract
We derive a general method for determining the necessary and sufficient conditions for exact factorization |Ψ=⊗_p |ψ_p 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.