Editing Polar decomposition (section)

==Properties==
The polar decomposition of the [[complex conjugate]] of <math>A</math> is given by <math>\overline{A} = \overline{U}\overline{P}.</math> Note that
<math display="block">
 \det A = \det U \det P = e^{i\theta} r
</math>
gives the corresponding polar decomposition of the [[determinant]] of ''A'', since <math>\det U = e^{i\theta},</math> and <math>\det P = r = |\det A|.</math> In particular, if <math>A</math> has determinant 1, then both <math>U</math> and <math>P</math> have determinant 1.

The positive-semidefinite matrix ''P'' is always unique, even if ''A'' is [[Singular matrices|singular]], and is denoted as
<math display="block">
 P = (A^* A)^{1/2},
</math>
where <math>A^*</math> denotes the [[conjugate transpose]] of <math>A</math>. The uniqueness of ''P'' ensures that this expression is well-defined. The uniqueness is guaranteed by the fact that <math>A^* A</math> is a positive-semidefinite Hermitian matrix and, therefore, has a unique positive-semidefinite Hermitian [[square root of a matrix|square root]].<ref>{{harvnb|Hall|2015|loc=Lemma 2.18}}.</ref> If ''A'' is invertible, then ''P'' is positive-definite, thus also invertible, and the matrix ''U'' is uniquely determined by
<math display="block">
 U = AP^{-1}.
</math>

===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.

===Relation to normal matrices===

The matrix <math>A</math> with polar decomposition <math>A = UP</math> is [[normal matrix|normal]] if and only if <math>U</math> and <math>P</math> [[Commuting matrices|commute]] (<math>UP = PU</math>), or equivalently, they are [[Diagonalizable matrix#Simultaneous diagonalization|simultaneously diagonalizable]].