Defective Hamiltonian matrix imaginary eigenvalues and losslessness

Abstract

The solutions to algebraic Riccati equation (ARE) have widespread applications in the area of control and network theory. For certain solutions of the Riccati equation, namely so-called ‘semi-stabilizing’ solutions, the corresponding Hamiltonian matrix has two or more purely imaginary eigenvalues. In this paper we explore the relationship between existence of such imaginary eigenvalues and lossless trajectories present in the system. It is known that under suitable conditions, such imaginary eigenvalues of the corresponding Hamiltonian matrix are ‘defective’, i.e., there are insufficient corresponding independent eigenvectors for the given eigenvalue. This poses theoretical and numerical difficulties in computing the solutions of the corresponding ARE. In this paper, we formulate conditions under which such imaginary eigenvalues of the Hamiltonian matrix are non-defective. As an extreme case of non-defectiveness, we first formulate conditions under which a Hamiltonian matrix is normal, i.e. the matrix commutes with its transpose. We also provide conditions under which imaginary eigenvalues of the Hamiltonian matrix are defective. Keywords: controllability, observability, defective eigenvalues, normal matrices, all-pass, diagonalizability