Editing Polar decomposition (section)

===Relation to the SVD===

In terms of the [[singular value decomposition]] (SVD) of <math>A</math>, <math>A = W\Sigma V^*</math>, one has
<math display="block">\begin{align}
 P &= V\Sigma V^*, \\
 U &= WV^*,
\end{align}</math>
where <math>U</math>, <math>V</math>, and <math>W</math> are unitary matrices ([[Orthogonal matrix|orthogonal]] if the field is the reals <math>\mathbb{R}</math>). This confirms that <math>P</math> is positive-definite, and <math>U</math> is unitary. Thus, the existence of the SVD is equivalent to the existence of polar decomposition.

One can also decompose <math>A</math> in the form
<math display="block">
 A = P'U.
</math>
Here <math>U</math> is the same as before, and <math>P'</math> is given by
<math display="block">
 P' = UPU^{-1} = (AA^*)^{1/2} = W \Sigma W^*.
</math>
This is known as the left polar decomposition, whereas the previous decomposition is known as the right polar decomposition. Left polar decomposition is also known as reverse polar decomposition.

The '''polar decomposition''' of a square invertible real matrix <math>A</math> is of the form
<math display="block">
 A = [A] R,
</math>
where <math>[A] \equiv \left(AA^\mathsf{T}\right)^{1/2}</math> is a [[positive-semidefinite matrix|positive-definite]] [[Hermitian matrix|matrix]], and <math>R = [A]^{-1} A</math> is an orthogonal matrix.