Cayley map for symplectic groups
A \(2n\!\times\!2n\) matrix \(g\) belongs to the symplectic group \(\mathrm{Sp}(2n,\mathbb{R}\)) if and only if \(g^tJg=J\), and a \(2n\!\times\!2n\) matrix \(X\) belongs to the corresponding Lie algebra \(sp(2n,\mathbb{R})\) if and only if \(X^tJ+JX=0\). The matrix \(J\equiv J_{2n}\) satisfying \(J^2=-I\) and \(J^t=-J\) is the symplectic unit. Since \(X^tJ+JX=0\ \Rightarrow \ (JX)^t=JX\), the Lie algebra of the symplectic group can be described by using the symmetric matrices as \(sp(2n,\mathbb{R})=\{\, JS\ |\ S^t=S\, \}\). By using this description and the Cayley map \(X\mapsto (I+X)/(I-X)\), the authors obtain an interesting parametrization for the elements of \(\mathrm{Sp}(2n,\mathbb{R}\)). They describe in terms of parameters the inverse of an element and the composition of two elements. Some numerical examples are also included. For the entire collection see [Zbl 1468.53002].
scientific article; zbMATH DE number 7391058
scientific article