Did you know that you can do a Löwdin orthogonalization by a singular value decomposition? Usually, when I hear Löwdin orthogonalization, I think of some weird S1/2 matrix, which scares me and I tend to stay away from it... But this pdf from the University of Oregon claims that you can do it in a different way. And it seems to work.
Say you have a matrix A and you want an orthogonal matrix that resembles it as closely as possible. What do you do? First you do a singular value decomposition of A:
Here U and V are orthogonal matrices and Λ is a diagonal matrix. We can now construct
which is an orthogonal matrix, since U and V are both orthogonal matrices. But even more, A' is the orthogonal matrix that best resembles A in the sense that for all orthogonal matrices Q it minimizes the distance with respect to the Frobenius norm
That is all you have to do.
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