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The question is quite old, but I can give a meaningful answer here. One of the most common algorithms to solve algebraic Riccati equations $A^TX+XA+Q=XGX$ (the Schur method) goes as follows: Compute a Schur decomposition of the Hamiltonian matrix $H = \begin{bmatrix}A& G\\ Q & -A^T \end{bmatrix}$ Reorder so that the eigenvalues in the left-half ...