Residue and monodromy of a constant Fuchsian system ($\Phi (z)=\exp (R\log z),M=\exp (2\pi iR)$)

A fundamental matrix for $F^{\prime }=RF/z$ on a chosen logarithm branch is $\exp (R\log z)$. A counterclockwise circuit changes the logarithm by $2\pi i$, giving the displayed monodromy matrix and the cyclic group it generates. Changing the solution basis conjugates the matrix. The residue eigenvalues exponentiate to the monodromy eigenvalues; the latter do not determine the former uniquely.