Algorithms and Properties for Positive Symmetrizable Matrices

Elisângela Silva Dias, Diane Castonguay, Mitre Costa Dourado


Matrices are the most common representations of graphs. They are also used for representing algebras and cluster algebras. This paper shows 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. Here, symmetrizable matrices are those which are symmetric when multiplied by a diagonal matrix with positive entries called symmetrizer matrix. We conjecture that this problem is NP-complete and we show that it is in NP by generalizing Sylvester's criterion for symmetrizable matrices. We straighten known coefficient limits for such matrices.


Symmetrizable matrix; positive quasi-Cartan matrix; algorithm.

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