  In this paper, we extend the eigenvalue method of the algebraic Riccati
equation to the differential Riccati equation (DRE) in contraction analysis.
One of the main results is showing that solutions to the DRE can be expressed
as functions of nonlinear eigenvectors of the differential Hamiltonian matrix.
Moreover, under an assumption for the differential Hamiltonian matrix, real
symmetricity, regularity, and positive semidefiniteness of solutions are
characterized by nonlinear eigenvalues and eigenvectors.
