**S**

^{1/2}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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