Fractional-linear invariance of the symplex-module algorithm for decomposition in multidimensional continued fractions

Using the simplex-module algorithm one can decompose real numbers $\alpha=(\alpha_1,\dots,\alpha_d)$ into multidimensional continued fractions. We verified the invariance of this algorithm under fractional-linear transformations $\alpha'=(\alpha'_1,\dots,\alpha'_d)=U\langle\alpha\rangle$ with matrices $U$ in the unimodular group $\mathrm{GL}_{d+1}(\mathbb Z)$, and prove the conservation of a linear recurrence and the approximation order for convergent fractions to the transformed $\alpha'$.