Galois groups and connection matrices of $q$-difference equations

We study the Galois group of a matrix $q$-difference equation with rational coefficients which is regular at $0$ and $\infty$, in the sense of (difference) Picard-Vessiot theory, and show that it coincides with the algebraic group generated by matrices $C(u)C(w)^{-1}$ $u,w\in \mathbb{C} ^*$, where $C(z)$ is the Birkhoff connection matrix of the equation.