Editing Orthogonal matrix (section)

===Nearest orthogonal matrix===

The problem of finding the orthogonal matrix {{mvar|Q}} nearest a given matrix {{mvar|M}} is related to the [[Orthogonal Procrustes problem]]. There are several different ways to get the unique solution, the simplest of which is taking the [[singular value decomposition]] of {{mvar|M}} and replacing the singular values with ones. Another method expresses the {{mvar|R}} explicitly but requires the use of a [[matrix square root]]:<ref>[http://people.csail.mit.edu/bkph/articles/Nearest_Orthonormal_Matrix.pdf "Finding the Nearest Orthonormal Matrix"], [[Berthold K.P. Horn]], [[MIT]].</ref>
<math display="block">Q = M \left(M^\mathrm{T} M\right)^{-\frac 1 2}</math>

This may be combined with the Babylonian method for extracting the square root of a matrix to give a recurrence which converges to an orthogonal matrix quadratically:
<math display="block">Q_{n + 1} = 2 M \left(Q_n^{-1} M + M^\mathrm{T} Q_n\right)^{-1}</math>
where {{math|1=''Q''<sub>0</sub> = ''M''}}.

These iterations are stable provided the [[condition number]] of {{mvar|M}} is less than three.<ref>[http://www.maths.manchester.ac.uk/~nareports/narep91.pdf "Newton's Method for the Matrix Square Root"] {{Webarchive|url=https://web.archive.org/web/20110929131330/http://www.maths.manchester.ac.uk/~nareports/narep91.pdf |date=2011-09-29 }}, Nicholas J. Higham, Mathematics of Computation, Volume 46, Number 174, 1986.</ref>

Using a first-order approximation of the inverse and the same initialization results in the modified iteration:

<math display="block">N_{n} = Q_n^\mathrm{T} Q_n</math>
<math display="block">P_{n} = \frac 1 2 Q_n N_{n}</math>
<math display="block">Q_{n + 1} = 2 Q_n + P_n N_n - 3 P_n</math>