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

TL;DR

This paper introduces a combinatorial approach using planar networks and cluster subalgebras to test and realize positive definite matrices, expanding the understanding of their structure and relationships.

Contribution

It provides a novel combinatorial realization of all matrices via planar networks and establishes a new test method for positive definiteness using $LDU$-decompositions and cluster subalgebras.

Findings

A combinatorial realization of all matrices through planar networks.

A new test method for positive definite matrices based on $LDU$-decompositions.

Established relationships between positive definite matrices and cluster subalgebras.

Abstract

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.