Symplectic decomposition from submatrix determinants

An important theorem in Gaussian quantum information tells us that we can diagonalise the covariance matrix of any state via a symplectic transformation. Whilst diagonal form is easy to find, process for finding diagonalising be more difficult, and common, existing method requires taking powers, which demanding analytically. Inspired by recently presented technique eigenvectors Hermitian from certain submatrix eigenvalues, derive similar determinants, could prove useful information.