Indefinite inner products: Conjugation

This chapter studies matrices (or linear transformations) that are selfadjoint or skewadjoint with respect to a nondegenerate hermitian or skewhermitian inner product. As an application of the canonical forms obtained in chapters 8 and 9, canonical forms for such matrices are derived in this chapter. Matrices that are skewadjoint with respect to skewhermitian inner products are known as Hamiltonian matrices; they play a key role in many applications such as linear control systems. The canonical forms reveal invariant Lagrangian subspaces; in particular, they give criteria for existence of such subspaces. Another application involves boundedness and stable boundedness of linear systems of differential equations with constant coefficients under suitable symmetry requirements.