New Parameterizations of \(\mathrm{SL}(2,\mathbb{R})\) and Some Explicit Formulas for Its Logarithm

Here we demonstrate some of the benefits of a novel parameterization of the Lie groups $\mathrm{Sp}(2,\mathbb{R}) ≅ \mathrm{SL}(2,\mathbb{R})$. Relying on the properties of the exponential map $\mathfrak{sl}(2,\mathbb{R}) \rightarrow \mathrm{SL}(2,\mathbb{R})$, we have found a few explicit formulas for the logarithm of the matrices in these groups. Additionally, the explicit analytic description of the ellipse representing their field of values is derived and this allows a direct graphical identification of various types.