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The trick is trying to find out why that matrix has real eigenvalues in the first place. Usually it is because a suitable set of conjugations turns it into a symmetric matrix, and then you can reduce to a symmetric computation. Multiplying and dividing by $(AC)^{-1}$ you can rewrite $$ D_1 = C^{-1}BAC (C^{-1}BAC+I)^{-1}, $$ so your computation is equivalent ...


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