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.

关键词： positive definite matrix,  generalized Jacobi matrix,  planar network,  double wiring diagram,  cluster subalgebras