Editing Polar decomposition (section)

=== General derivation ===
The SVD of a square matrix <math>A</math> reads <math>A = W D^{1/2} V^*</math>, with <math>W, V</math> unitary matrices, and <math>D</math> a diagonal, positive semi-definite matrix. By simply inserting an additional pair of <math>W</math>s or <math>V</math>s, we obtain the two forms of the polar decomposition of <math>A</math>:<math display="block">
  A = WD^{1/2}V^* =
  \underbrace{\left(W D^{1/2} W^*\right)}_P \underbrace{\left(W V^*\right)}_U =
  \underbrace{\left(W V^*\right)}_U \underbrace{\left(VD^{1/2} V^*\right)}_{P'}.
</math>More generally, if <math> A </math> is some rectangular <math>
  n\times m </math> matrix, its SVD can be written as <math> A=WD^{1/2}V^* </math> where now <math> W </math> and <math> V </math> are isometries with dimensions <math> n\times r </math> and <math>
 m\times r </math>, respectively, where <math> r\equiv\operatorname{rank}(A) </math>, and <math> D </math> is again a diagonal positive semi-definite square matrix with dimensions <math> r\times r </math>. We can now apply the same reasoning used in the above equation to write <math> A=PU=UP'</math>, but now <math> U\equiv WV^* </math> is not in general unitary. Nonetheless, <math> U </math> has the same support and range as <math> A </math>, and it satisfies <math> U^* U=VV^* </math> and <math> UU^*=WW^* 
</math>. This makes <math> U </math> into an isometry when its action is restricted onto the support of <math> A </math>, that is, it means that <math> U </math> is a [[partial isometry]].

As an explicit example of this more general case, consider the SVD of the following matrix:<math display="block">
  A\equiv \begin{pmatrix}1&1\\2&-2\\0&0\end{pmatrix} =
\underbrace{\begin{pmatrix}1&0\\0&1\\0&0\end{pmatrix}}_{\equiv W}
\underbrace{\begin{pmatrix}\sqrt2&0\\0&\sqrt8\end{pmatrix}}_{\sqrt D}
\underbrace{\begin{pmatrix}\frac1{\sqrt2} & \frac1{\sqrt2} \\ \frac1{\sqrt2} & -\frac1{\sqrt2}\end{pmatrix}}_{V^\dagger}. 
</math>We then have<math display="block">
  WV^\dagger = \frac1{\sqrt2}\begin{pmatrix}1&1 \\ 1&-1 \\ 0&0\end{pmatrix} 
</math>which is an isometry, but not unitary. On the other hand, if we consider the decomposition of<math display="block">
  A\equiv \begin{pmatrix}1&0&0\\0&2&0\end{pmatrix} =
\begin{pmatrix}1&0\\0&1\end{pmatrix}
\begin{pmatrix}1&0\\0&2\end{pmatrix}
\begin{pmatrix}1&0&0\\0&1&0\end{pmatrix}, 
</math>we find<math display="block">
  WV^\dagger =\begin{pmatrix}1&0&0\\0&1&0\end{pmatrix},  
</math>which is a partial isometry (but not an isometry).