On Fillmore's theorem over integrally closed domains
Alexander Stasinski
TL;DR
The paper investigates diagonalization analogues of Fillmore's theorem over rings. It shows that the Borobia–Tan generalisation holds for integrally closed domains by leveraging the rational canonical form, but provides a counterexample demonstrating that integral closedness is essential. It then proves that over a PID with $n\ge 3$, a natural necessary condition identified by Tan is in fact sufficient for $R$-similarity to a diagonal matrix, using a Laffey–Reams-type normal form and existing lemmas. The results delineate the limits of generalising Fillmore's theorem to ring settings, highlighting a 2x2 obstruction and leaving open the question for integrally closed domains that are not PIDs.
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 $γ_{1},\dots,γ_{n}\in K$ are such that $γ_{1}+\dots+γ_{n}=\operatorname{Tr}(A)$, then $A$ is $K$-similar to a matrix with diagonal $(γ_{1},\dots,γ_{n})$. 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 $(γ_{1},\dots,γ_{n})$. 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 $(γ_{1},\dots,γ_{n})$. We show that when $R$ is a PID and $n\geq3$, Tan's condition is also sufficient.