Editing Singular value decomposition (section)

===Nearest orthogonal matrix===
It is possible to use the SVD of a square matrix {{tmath|\mathbf A}} to determine the [[orthogonal matrix]] {{tmath|\mathbf O}} closest to {{tmath|\mathbf A.}} The closeness of fit is measured by the [[Frobenius norm]] of {{tmath|\mathbf O - \mathbf A.}} The solution is the product {{tmath|\mathbf U \mathbf V^*.}}<ref>[http://www.wou.edu/~beavers/Talks/Willamette1106.pdf The Singular Value Decomposition in Symmetric (Lowdin) Orthogonalization and Data Compression]</ref> This intuitively makes sense because an orthogonal matrix would have the decomposition {{tmath|\mathbf U \mathbf I \mathbf V^*}} where {{tmath|\mathbf I}} is the identity matrix, so that if {{tmath|\mathbf A {{=}} \mathbf U \mathbf \Sigma \mathbf V^*}} then the product {{tmath|\mathbf A {{=}} \mathbf U \mathbf V^*}} amounts to replacing the singular values with ones.  Equivalently, the solution is the unitary matrix {{tmath|\mathbf R {{=}} \mathbf U \mathbf V^*}} of the Polar Decomposition <math>\mathbf M = \mathbf R \mathbf P = \mathbf P' \mathbf R</math> in either order of stretch and rotation, as described above.

A similar problem, with interesting applications in [[shape analysis (digital geometry)|shape analysis]], is the [[orthogonal Procrustes problem]], which consists of finding an orthogonal matrix {{tmath|\mathbf O}} which most closely maps {{tmath|\mathbf A}} to {{tmath|\mathbf B.}} Specifically,

<math display=block>
\mathbf{O} = \underset\Omega\operatorname{argmin} \|\mathbf{A}\boldsymbol{\Omega} - \mathbf{B}\|_F \quad\text{subject to}\quad \boldsymbol{\Omega}^\operatorname{T}\boldsymbol{\Omega} = \mathbf{I},
</math>

where <math>\| \cdot \|_F</math> denotes the Frobenius norm.

This problem is equivalent to finding the nearest orthogonal matrix to a given matrix <math>\mathbf M = \mathbf A^\operatorname{T} \mathbf B</math>.