Left-eigenvectors are certificates of the Orbit Problem

TL;DR

This paper links the Orbit Problem to eigenvector-based certificates, providing reductions and proofs of existence for integer-coefficient transformations, advancing understanding of reachability in linear systems.

Contribution

It introduces a reduction of the Orbit Problem to a generalized eigenvector problem and proves the existence of certificates for integer-coefficient transformations.

Findings

Certificates exist for integer-coefficient transformations

Reduction to generalized eigenvector problem

Connection between Orbit Problem and eigenvector computation

Abstract

This paper investigates the connexion between the Kannan-Lipton Orbit Problem and the polynomial invariant generator algorithm PILA based on eigenvectors computation. Namely, we reduce the problem of generating linear and polynomial certificates of non-reachability for the Orbit Problem for linear transformations with rational coefficients to the generalized eigenvector problem. Also, we prove the existence of such certificates for any transformation with integer coefficients, which is not the case with rational coefficients.