The equation XDX + XA + A^*X - C = 0 is studied under the assumptions that D is positive semidefinite and to pure imaginary eigenvalues of the Hamiltonian matrix pmatrix A & D \\ C & - A^* pmatrix correspond only elementary divisors of even degree. A necessary and sufficient condition on the existence and uniqueness of hermitian solutions is given. The approach is based on symplectic transformations of Hamiltonian matrices.
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Hȧrald K. Wimmer (1982) studied this question.
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