The realization of positive definite matrices via planar networks and mixing-type sub-cluster algebras

As an improvement of the combinatorial realization of totally positive matrices via the essential positive weightings of certain planar network by S.Fomin and A.Zelevinsky[7],in this paper,we give a test method of positive definite matrices via the planar networks and the so-called mixing-type sub-cluster algebras respectively,introduced here originally.This work firstly gives a combinatorial realization of all matrices through planar network,and then sets up a test method for positive definite matrices by LDU-decompositions and the horizontal weightings of all lines in their planar networks.On the other hand,mainly the relationship is built between positive definite matrices and mixing-type sub-cluster algebras.