R-Orbit Reflexive Operators

TL;DR

This paper characterizes orbit reflexivity and R-orbit reflexivity for real matrices, revealing differences from the complex case and linking properties to linear dependence over Q.

Contribution

It provides a complete characterization of orbit reflexivity for real matrices, contrasting with the complex case and introducing new criteria based on linear dependence over Q.

Findings

Real matrix orbit reflexivity depends on linear dependence over Q.

Every n-by-n matrix over an uncountable field is algebraically F-orbit reflexive.

Differences between real and complex cases are established.

Abstract

We completely characterize orbit reflexivity and R-orbit reflexivity for square matrices over the real numbers. Unlike the complex case in which every matrix is orbit reflexive and C-orbit reflexivity is characterized solely in terms of the Jordan form, the orbit reflexivity and R-orbit reflexivity of a real matrix is described in terms of the linear dependence over Q of certain elements of R/Q. We also show that every n-by-n matrix over an uncountable field F is algebraically F-orbit reflexive.