On Fillmore's theorem over integrally closed domains

TL;DR

This paper extends Fillmore's theorem to integrally closed domains, showing the conditions under which matrices can be similar to diagonal matrices with prescribed entries, and provides counterexamples and sufficiency results.

Contribution

It generalizes Fillmore's theorem from fields to integrally closed domains and establishes sufficiency of Tan's necessary condition for PIDs with dimension at least 3.

Findings

Extension of Fillmore's theorem to integrally closed domains.

Counterexample showing the limit of generalization beyond integrally closed domains.

Sufficiency of Tan's condition for PIDs with matrix size n≥3.

Abstract

A well-known theorem of Fillmore says that if $A\in\operatorname{M}_{n}(K)$ is a non-scalar matrix over a field $K$ and $\gamma_{1},\dots,\gamma_{n}\in K$ are such that $\gamma_{1}+\dots+\gamma_{n}=\operatorname{Tr}(A)$, then $A$ is $K$-similar to a matrix with diagonal $(\gamma_{1},\dots,\gamma_{n})$. Building on work of Borobia, Tan extended this by proving that if $R$ is a unique factorisation domain with field of fractions $K$ and $A\in\operatorname{M}_{n}(R)$ is non-scalar, then $A$ is $K$-similar to a matrix in $\operatorname{M}_{n}(R)$ with diagonal $(\gamma_{1},\dots,\gamma_{n})$. We note that Tan's argument actually works when $R$ is any integrally closed domain and show that the result cannot be generalised further by giving an example of a matrix over a non-integrally closed domain for which the result fails. Moreover, Tan gave a necessary condition for $A\in\operatorname{M}_{n}(R)$ to be $R$-similar to a matrix with diagonal $(\gamma_{1},\dots,\gamma_{n})$. We show that when $R$ is a PID and $n\geq3$, Tan's condition is also sufficient.