Positive determinacy of h-Shuhan matrices with $h<2$

TL;DR

This paper introduces h-Shuhan matrices, generalizes Cartan matrices, and analyzes their eigenvalues, showing they are positive definite for h<2 and providing bounds on their largest eigenvalues.

Contribution

It defines h-Shuhan matrices as a generalization of Cartan matrices and characterizes their positive semi-definite cases for h<2.

Findings

h-Shuhan matrices are positive semi-definite for h<2

Largest eigenvalue of $\hat{B}_{n}^{h}$ is less than h plus approximately 2.05

Eigenvalue increases with n but remains bounded

Abstract

In this paper, we define h-Shuhan matrix, which is the generalization of the generalized Cartan matrix, and find the h-Shuhan matrices for all positive semi-definite ( or generalized positive semi-definite, virtual positive semi-definite) with $h<2$. Furthermore, we know that the largest eigenvalue of the matrix $\hat{B}_{n}^{h}$ increases with $n$, but it is always less than $h$ plus a constant $\epsilon\approx2.04998$.