Equations of the form \(X\bar X = A\) with skew-Hamiltonian matrices \(A\) (Q1761031)

For \(H = i \left( \begin{matrix} O_n & I_n\\ -I_n& O_n \end{matrix} \right) = i J\), a matrix \(B \in C^{2n,2n}\) is \(H\)-Hermitian if \(HB = B^*H\). Likewise \(A \in C^{2n,2n}\) is called skew-Hamiltonian if \(JA = A^TJ\). The paper proves that for each \(H\)-Hermitian matrix \(B\) the matrix \(A = B \overline{B}\) is skew-Hamiltonian and that for any skew-Hamiltonian matrix \(A\) that is similar to a real matrix without any real negative eigenvalues (phrased via the eigen- and Jordan structure of \(A\) as condition (a) in the paper) the equation \(X\overline{X} = A\) has an \(H\)-Hermitian matrix solution \(X\).