Algorithms and Properties for Positive Symmetrizable Matrices

Abstract

ABSTRACT. Matrices are one of the most common representations of graphs. They are also used for representing algebras and cluster algebras. A symmetrizable matrix M is one for which there is a diagonal matrix D with positive entries, called symmetrizer matrix, such that DM is symmetric. This paper provides some properties of matrices in order to facilitate the understanding of symmetrizable matrices with specific characteristics, called positive quasi-Cartan companion matrices, and the problem of localizing them. We sharpen known coefficient limits for such matrices. By generalizing Sylvester's criterion for symmetrizable matrices we show that the localization problem is in NP and conjectured that it is NP-complete.

Multiequilibria Analysis for a Class of Collective Decision-Making Networked Systems

“…A matrix A ∈ R n×n is (diagonally) symmetrizable if DA is symmetric for some diagonal matrix D with positive diagonal entries [27]. The matrix DA is called symmetrization of A and the matrix D is called symmetrizer of A [28]. The eigenvalues of a symmetrizable matrix are real.…”

Investigating mixed-sign equilibria for nonlinear collective decision-making systems

“…The matrix DA is called symmetrization of A and the matrix D is called symmetrizer of A. The eigenvalues of a symmetrizable matrix are real [20], [21], [22]. A ∈ R n×n is symmetrizable if and only if it is sign symmetric, i.e.…”