ABSTRACT
Matrices are one of the most common representations of graphs. They are also used forrepresenting 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.
Keywords:
symmetrizable matrix; positive quasi-Cartan matrix; algorithm
