Versal deformations and normal forms for reversible and Hamiltonian linear systems
In relation to linear differential equations \(\dot x = Ax\), the author studies the adjoint action of certain subgroups \({\mathcal G}\) of the (real) general linear group \(GL(m)\) on certain subspaces \({\mathbf u}\) of the Lie algebra \({\mathbf g} {\mathbf l} (m)\). Three cases, in each of which \(m\) is even, are considered. (1) Let \(R\) be an involution similar to \(- R\) and call \(A\) \(R\)-reversible if \(AR + RA = 0\). Then \({\mathbf u}_1\) is the set of \(R\)-reversible matrices and \({\mathcal G}_1\) the set of nonsingular matrices commuting with \(R\). (2) \({\mathbf u}_2\) is the set of Hamiltonian matrices and \({\mathcal G}_2\) the corresponding symplectic group, i.e. \({\mathbf u}_2 = {\mathbf s} {\mathbf p} (m)\), \({\mathcal G}_2 = Sp(m)\). (3) \({\mathbf u}_3 = {\mathbf u}_1 \frown {\mathbf u}_2\), \({\mathcal G}_3 = {\mathcal G}_1 \frown {\mathcal G}_2\). The author determines normal forms for the action of \({\mathcal G}\) on \({\mathbf u}\) and compares them with the normal forms for the action of \(GL (m)\) on \({\mathbf g} {\mathbf l} (m)\). The task here is considerably simplified by adroit use of the Jordan-Chevalley decomposition \(A = S + N\) to reduce the problem to the much easier semisimple case. The author also determines in each case a corresponding versal deformation. This means constructing at a given \(A \in {\mathbf u}\) a section in \({\mathbf u}\) transverse to the \({\mathcal G}\)-orbit of \(A\). This is done by taking the tangent space to the orbit at \(A\) and forming its orthogonal complement with respect to the metric on \({\mathbf u}\) given by \((B,C) = \text{trace} (B^tC)\). The detailed results are set out in extensive tables at the end of the paper.
scientific article; zbMATH DE number 880563