Positive definite matrices in the view-points of planar networks and cluster subalgebras

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.Zelevisky \cite{[4]}, in this paper, we give the test method of positive definite matrices via the planar networks and the so-called cluster subalgebra 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 cluster subalgebras.