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Say you have a matrix you want to diagonalize, and you get the eigenvectors $\vec{\alpha}_i$ and arrange them to get the similarity matrix $P = (\vec{\alpha}_1, \vec{\alpha}_2,....)$. Say P is orthogonal and you want to restrict it to be 'special' (det $= 1$). Typically I would do this by normalizing the eigenvectors. When you do this though you will get a sign ambiguity in your normalization which you have to choose. How do you systematically do this so as to reject the ' improper' rotations? Or is there a better way than choosing the normalization of the eigenvectors to get the 'proper' rotations?

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Typically, you will normalize the eigenvectors in any way you please, and just be careful to pick the right sign when you come to the last eigenvector.

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