Solvability of Matrix-Exponential Equations

TL;DR

This paper investigates the solvability of matrix-exponential equations, proving undecidability in general but decidability when matrices commute, with implications for hybrid automata reachability and using advanced algebraic and transcendental number theory.

Contribution

It establishes the decidability boundary for matrix-exponential equations, providing new theoretical tools and results, including a novel uniqueness theorem for matrix logarithms.

Findings

Undecidability of the general problem via reduction from Hilbert's Tenth Problem.

Decidability when matrices commute, using algebraic and transcendental number theory.

Introduction of a new result on the uniqueness of matrix logarithms.

Abstract

We consider a continuous analogue of Babai et al.'s and Cai et al.'s problem of solving multiplicative matrix equations. Given $k+1$ square matrices $A_{1}, \ldots, A_{k}, C$, all of the same dimension, whose entries are real algebraic, we examine the problem of deciding whether there exist non-negative reals $t_{1}, \ldots, t_{k}$ such that \begin{align*} \prod \limits_{i=1}^{k} \exp(A_{i} t_{i}) = C . \end{align*} We show that this problem is undecidable in general, but decidable under the assumption that the matrices $A_{1}, \ldots, A_{k}$ commute. Our results have applications to reachability problems for linear hybrid automata. Our decidability proof relies on a number of theorems from algebraic and transcendental number theory, most notably those of Baker, Kronecker, Lindemann, and Masser, as well as some useful geometric and linear-algebraic results, including the Minkowski-Weyl theorem and a new (to the best of our knowledge) result about the uniqueness of strictly upper triangular matrix logarithms of upper unitriangular matrices. On the other hand, our undecidability result is shown by reduction from Hilbert's Tenth Problem.