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

We determine the canonical form of a Hamiltonian matrix X∈ sp(2 n, R) under symplectic similarity, and the canonical form of a matrix Y∈ 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. © 2017, Sociedad Matemática Mexicana.

Adjoint action; Canonical forms; Classical Lie groups; Cyclic subspaces; Eigenvalue problem; Hamiltonian matrix; Jordan canonical form

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