Non-negative integral matrices with given spectral radius and controlled dimension

TL;DR

This paper proves the existence of integral irreducible matrices with a given Perron number as spectral radius, with bounds on their dimension based on algebraic and arithmetic properties, linking matrix theory and number theory.

Contribution

It establishes bounds on the dimension of integral irreducible matrices with a specified Perron spectral radius, connecting algebraic number theory and matrix theory.

Findings

Existence of integral irreducible matrices with prescribed Perron spectral radius.

Dimension bounds depend on algebraic degree and Galois conjugates.

Correspondence with shifts of finite type and entropy.

Abstract

A celebrated theorem of Lind states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number $p$, we prove that there is an integral irreducible matrix with spectral radius $p$, and with dimension bounded above in terms of the algebraic degree, the ratio of the first two largest Galois conjugates, and arithmetic information about the ring of integers of its number field. This arithmetic information can be taken to be either the discriminant or the minimal Hermite-like thickness. Equivalently, given a Perron number $p$, there is an irreducible shift of finite type with entropy $\log(p)$ defined as an edge shift on a graph whose number of vertices is bounded above in terms of the aforementioned data.