Algorithms and Properties for Positive Symmetrizable Matrices

TL;DR

This paper investigates properties of symmetrizable matrices, introduces algorithms to identify and construct positive quasi-Cartan companions, and proves the NP-completeness of the decision problem for such matrices.

Contribution

It presents four algorithms for analyzing symmetrizable matrices and establishes the NP complexity of finding positive quasi-Cartan companions.

Findings

Developed algorithms to determine symmetrizer matrices and positivity

Proved the problem of finding positive quasi-Cartan companions is NP-complete

Facilitated understanding of matrix properties related to graph and algebra representations

Abstract

Matrices are the most common representations of graphs. They are also used for the representation of algebras and cluster algebras. This paper shows some properties of matrices in order to facilitate the understanding and locating symmetrizable matrices with specific characteristics, called positive quasi-Cartan companion matrices. Here, symmetrizable matrix are those which are symmetric when multiplied by a diagonal matrix with positive entries called symmetrizer matrix. Four algorithms are developed: one to decide whether there is a symmetrizer matrix; second to find such symmetrizer matrix; another to decide whether the matrix is positive or not; and the last to find a positive quasi-Cartan companion matrix, if there exists. The third algorithm is used to prove that the problem to decide if a matrix has a positive quasi-Cartan companion is NP.