Fractional-linear invariance of multidimensional continued fractions

With the help of the simplex-karyon algorithm it is possible to decompose real numbers $\alpha=(\alpha_1,\dots,\alpha_d)$ into multidimensional continued fractions. We prove the invariance of this algorithm under fractional-linear transformations $\alpha'=(\alpha'_1,\dots,\alpha'_d)=U\langle\alpha\rangle$ with matrices $U$ from the unimodular group $\mathrm{GL}_{d+1}(\mathbb Z)$. The best convergent fractions of the transformed $\alpha'$ are found.