Periodic Coxeter matrices and their associated quadratic forms

We study relations between the periodicity of a Coxeter matrix and the positivity or the non-negativity of a quadratic form with respect to a Cartan matrix of an artinian ring. We introduce the notion of the weak periodicity of a Coxeter matrix induced by a non-negative quadratic form and we characterize the weak periodicity condition and the periodicity condition in terms of eigenvalues of a Coxeter matrix.