Real canonical forms for the adjoint action of the Lie groups $${\text {Sp}}(V,B)$$ Sp ( V , B ) and $${\text {O}}(V,B)$$ O ( V , B ) on their Lie algebras

Abstract

We determine the canonical form of a Hamiltonian matrix $$X\in \mathfrak {sp}(2n,\mathbb {R})$$ under symplectic similarity, and the canonical form of a matrix $$Y\in \mathfrak {o}(m)$$ in the orthogonal Lie algebra under similarity. This is a well known problem, and it has been solved by means of different techniques. Our contribution is to provide a new solution through elementary linear algebra. As an application, a list of the non-equivalent two- and four-dimensional quadratic Hamiltonians is given.