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In [[mathematics]], particularly in [[linear algebra]] and [[functional analysis]], the '''polar decomposition''' of a [[matrix (mathematics)|matrix]] or [[linear operator]] is a [[matrix decomposition|factorization]] analogous to the [[Complex number#Polar form|polar form]] of a nonzero [[complex number]] ''z'' as
<math>z = r e^{i \theta}\,</math>
where ''r'' is the [[absolute value]] of ''z'' (a [[Positive number|positive]] [[real number]]), and <math>e^{i\theta}</math> is called the complex sign of ''z''.


==Matrix polar decomposition==
The '''polar decomposition''' of a square complex matrix ''A'' is a [[matrix decomposition]] of the form
:<math>A = UP\,</math>
where ''U'' is a [[unitary matrix]] and ''P'' is a [[positive-semidefinite matrix|positive-semidefinite]] [[Hermitian matrix]]. Intuitively, the polar decomposition separates ''A'' into a component that stretches the space along a set of orthogonal axes, represented by ''P'', and a rotation represented by ''U''. The decomposition of the [[complex conjugate]] of <math>A</math> is given by <math>\overline{A} = \overline{U}</math> <math>\overline{P}</math>.

This decomposition always exists; and so long as ''A'' is [[invertible matrix|invertible]], it is unique, with ''P'' [[positive-definite matrix|positive-definite]]. Note that
:<math>\det A = \det U\,\det P = re^{i\theta}</math>
gives the corresponding polar decomposition of the [[determinant]] of ''A'', since <math>\det P = r = |\det A|</math> and <math>\det U = e^{i\theta}</math>.

The matrix ''P'' is always unique, even if ''A'' is singular, and given by
:<math>P = \sqrt{A^*A}</math>
where ''A''* denotes the [[conjugate transpose]] of ''A''. This expression is meaningful since a positive-semidefinite Hermitian matrix has a unique positive-semidefinite [[Square root of a matrix|square root]]. If ''A'' is invertible, then the matrix ''U'' is given by
:<math>U = AP^{-1}.\,</math>

In terms of the [[singular value decomposition]] of ''A'', ''A = W Σ V<sup>*''</sup>, one has
:<math>P = V \Sigma V^*\,</math>
:<math>U = W V^*\,</math>
confirming that ''P'' is positive-definite and ''U'' is unitary.

One can also decompose ''A'' in the form
:<math>A = P'U\,</math>
Here ''U'' is the same as before and ''P''&prime; is given by
:<math>P' = UPU^{-1} = \sqrt{AA^*} = 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 matrix ''A'' is [[normal matrix|normal]] if and only if ''P''&prime; = ''P''. Then ''UΣ = ΣU'', and it is possible to diagonalise ''U'' with a unitary similarity matrix ''S'' that commutes with ''Σ'', giving ''S U S*'' = ''Φ<sup>−1</sup>'', where ''Φ'' is a diagonal unitary matrix of phases ''e<sup>iφ</sup>''. Putting ''Q = V S<sup>*</sup>'', one can then re-write the polar decomposition as

:<math> A = (Q \Phi Q^*)(Q \Sigma Q^*),\,</math>

so ''A'' then thus also has a [[spectral decomposition]]
:<math> A = Q  \Lambda Q^* \,</math>
with complex eigenvalues such that ''ΛΛ<sup>*</sup> = Σ<sup>2</sup>'' and a unitary matrix of complex eigenvectors ''Q''.

==Bounded operators on Hilbert space==
The '''polar decomposition''' of any [[bounded linear operator]] ''A'' between complex [[Hilbert space]]s is a canonical factorization as the product of a [[partial isometry]] and a non-negative operator.

The polar decomposition for matrices generalizes as follows: if ''A'' is a bounded linear operator then there is a unique factorization of ''A'' as a product ''A'' = ''UP'' where ''U'' is a partial isometry, ''P'' is a non-negative self-adjoint operator and the initial space of ''U'' is the closure of the range of ''P''.

The operator ''U'' must be weakened to a partial isometry, rather than unitary, because of the following issues. If ''A'' is the [[shift operator|one-sided shift]] on ''l''<sup>2</sup>('''N'''), then |''A''| = {''A*A''}<sup>½</sup> = ''I''. So if ''A'' = ''U'' |''A''|, ''U'' must be ''A'', which is not unitary.

The existence of a polar decomposition is a consequence of [[Douglas' lemma]]:

:'''Lemma''' If ''A'', ''B'' are bounded operators on a Hilbert space ''H'', and ''A*A'' &le; ''B*B'', then there exists a contraction ''C'' such that ''A = CB''. Furthermore, ''C'' is unique if ''Ker''(''B*'') &sub; ''Ker''(''C'').

The operator ''C'' can be defined by ''C(Bh)'' := ''Ah'' for all ''h'' in ''H'', extended by continuity to the closure of ''Ran''(''B''), and by zero on the orthogonal complement to all of ''H''. The lemma then follows since ''A*A'' ≤ ''B*B'' implies ''Ker''(''A'') ⊂ ''Ker''(''B'').

In particular. If ''A*A'' = ''B*B'', then ''C'' is a partial isometry, which is unique if ''Ker''(''B*'') ⊂ ''Ker''(''C'').
In general, for any bounded operator ''A'',

:<math>A^*A = (A^*A)^{\frac{1}{2}} (A^*A)^{\frac{1}{2}},</math>

where (''A*A'')<sup>½</sup> is the unique positive square root of ''A*A'' given by the usual [[functional calculus]]. So by the lemma, we have

:<math>A = U (A^*A)^{\frac{1}{2}}</math>

for some partial isometry ''U'', which is unique if ''Ker''(''A*'') ⊂ ''Ker''(''U''). Take ''P'' to be (''A*A'')<sup>½</sup> and one obtains the polar decomposition ''A'' = ''UP''. Notice that an analogous argument can be used to show ''A = P'U' '', where ''P' '' is positive and ''U' '' a partial isometry.

When ''H'' is finite dimensional, ''U'' can be extended to a unitary operator; this is not true in general (see example above). Alternatively, the polar decomposition can be shown using the operator version of [[singular value decomposition#Bounded operators on Hilbert spaces|singular value decomposition]].

By property of the [[continuous functional calculus]], ''|A|'' is in the [[C*-algebra]] generated by ''A''. A similar but weaker statement holds for the partial isometry: ''U'' is in the [[von Neumann algebra]] generated by ''A''. If ''A'' is invertible, the polar part ''U'' will be in the [[C*-algebra]] as well.

==Unbounded operators==
If ''A'' is a closed, densely defined [[unbounded operator]] between complex Hilbert spaces then it still has a (unique) '''polar decomposition'''

:<math>A = U |A|\,</math>

where |''A''| is a (possibly unbounded) non-negative self adjoint operator with the same domain as ''A'', and ''U'' is a partial isometry vanishing on the orthogonal complement of the range ''Ran''(|''A''|).

The proof uses the same lemma as above, which goes through for unbounded operators in general. If ''Dom''(''A*A'') = 
''Dom''(''B*B'') and ''A*Ah'' = ''B*Bh'' for all ''h'' ∈ ''Dom''(''A*A''), then there exists a partial isometry ''U'' such that ''A'' = ''UB''. ''U'' is unique if ''Ran''(''B'')<sup>⊥</sup>⊂ ''Ker''(''U''). The operator ''A'' being closed and densely defined ensures that the operator ''A*A'' is self-adjoint (with dense domain) and therefore allows one to define (''A*A'')<sup>½</sup>. Applying the lemma gives polar decomposition.

If an unbounded operator ''A'' is [[affiliated operator|affiliated]] to a von Neumann algebra '''M''', and ''A'' = ''UP'' is its polar decomposition, then ''U'' is in '''M''' and so is the spectral projection of ''P'', 1<sub>''B''</sub>(''P''), for any Borel set ''B'' in [0, ∞).

==Quaternion polar decomposition==
The polar decomposition of [[quaternion]]s '''H''' depends on the sphere <math>\lbrace x i + y j + z k \in H : x^2 + y^2 +z^2 = 1 \rbrace </math> of [[quaternion#Square roots of −1|square roots of minus one]]. Given any ''r'' on this sphere, and an angle –π < ''a'' ≤ π, the [[versor]] <math>e^{ar} = \cos (a) + r\ \sin (a) </math> is on the [[3-sphere]] of '''H'''.
For ''a'' = 0 and ''a'' = π, the versor is 1 or &minus;1 regardless of which ''r'' is selected.
The [[norm (mathematics)|norm]] ''t'' of a quaternion ''q'' is the [[Euclidean distance]] from the origin to ''q''.
When a quaternion is not just a real number, then there is a ''unique'' polar decomposition
<math>q = t e^{ar} \! .</math>

==Alternative planar decompositions==
In the [[Cartesian plane]], alternative planar [[ring (mathematics)|ring]] decompositions arise as follows:
* If ''x'' ≠ 0, ''z'' = ''x'' ( 1 + (''y''/''x'') ε) is a polar decomposition of a [[dual number]] ''z'' = ''x'' + ''y'' ε, where ε<sup>2</sup> = 0, ''i.e.'' ε is [[nilpotent]]. In this polar decomposition, the unit circle has been replaced by the line ''x'' = 1, the polar angle by the [[slope]] ''y/x'', and the radius ''x'' is negative in the left half-plane.

* If ''x''<sup>2</sup>&nbsp;≠&nbsp;''y''<sup>2</sup>, then the [[unit hyperbola]] ''x''<sup>2</sup>&nbsp;&minus;&nbsp;''y''<sup>2</sub>&nbsp;=&nbsp;1 and its conjugate ''x''<sup>2</sup>&nbsp;&minus;&nbsp;''y''<sup>2</sub>&nbsp;=&nbsp;&minus;1 can be used to form a polar decomposition based on the branch of the unit hyperbola through (1,0). This branch is parametrized by the [[hyperbolic angle]] ''a'' and is written
::<math>\cosh (a)  + j \ \sinh (a)  = \exp(a j) = e^{a j}</math>
:where ''j'' <sup>2</sup>&nbsp;=&nbsp;+1 and the arithmetic<ref>Sobczyk, G.(1995) "Hyperbolic Number Plane", ''[[College Mathematics Journal]] 26:268&ndash;80</ref> of [[split-complex number]]s is used. The branch through (&minus;1,0) is traced by &minus;''e''<sup>''a j''</sup>. Since the operation of multiplying by j reflects a point across the line ''y''&nbsp;=&nbsp;''x'', the second hyperbola has branches traced by ''je''<sup>''a j''</sup> or &minus;''je''<sup>''a j''</sup>. Therefore a point in one of the quadrants has a polar decomposition in one of the forms:
::<math>r e^{a j}, - r e^ {a j}, r j e^{a j}, - r j e^{a j}, \quad r > 0</math>
:The set { 1, &minus;1, j, &minus;j } has products that make it isomorphic to the [[Klein four-group]]. Evidently polar decomposition in this case involves an element from that group.

==Numerical determination of the matrix polar decomposition==
To compute an approximation of the polar decomposition ''A''=''UP'', usually the unitary factor ''U'' is approximated.<ref name="higham1986" /><ref name="byers2008" /> The iteration is based on [[Heron's method]] for the square root of ''1'' and computes, starting from <math>U_0=A</math>, the sequence
:<math>U_{k+1}=\frac12\left(U_k+(U_k^*)^{-1}\right)</math>, ''k''=0,1,2,...
The combination of inversion and Hermite conjugation is chosen so that in the singular value decomposition, the unitary factors remain the same and the iteration reduces to Heron's method on the singular values.

This basic iteration may be refined to speed up the process:

* Every step or in regular intervals, the range of the singular values of <math>U_k</math> is estimated and then the matrix is rescaled to <math>\gamma_kU_k</math> to center the singular values around ''1''. The scaling factor <math>\gamma_k</math> is computed using matrix norms of the matrix and its inverse. Examples of such scale estimates are:
:::<math>
  \gamma_k=\sqrt[4\;]{\frac{
    \|U_k^{-1}\|_1\,\|U_k^{-1}\|_\infty
     }{
    \|U_k\|_1\,\|U_k\|_\infty
   } }
</math>
:using the row-sum and column-sum [[Induced norm|matrix norms]] or
:::<math>
  \gamma_k=\sqrt{\frac{\|U_k^{-1}\|_F}{\|U_k\|_F}}
</math>
:using the [[Frobenius norm]]. Including the scale factor, the iteration is now
:<math>U_{k+1}=\frac12\left(\gamma_kU_k+\frac1{\gamma_k}(U_k^*)^{-1}\right)</math>, ''k''=0,1,2,...

* The [[QR decomposition]] can be used in a preparation step to reduce a singular matrix ''A'' to a smaller regular matrix, and inside every step to speed up the computation of the inverse.

* Heron' method for computing roots of <math>x^2-1=0</math> can be replaced by higher order methods, for instance based on [[Halley's method]] of third order, resulting in
:::<math>U_{k+1}=U_k\left(I+3\,U_k^*U_k\right)^{-1}\left(3\,I+U_k^*U_k\right)</math>, ''k''=0,1,2,...
:This iteration can again be combined with rescaling. This particular formula has the benefit that it also applicable to singular or rectangular matrices ''A''.

==See also==
*[[Cartan decomposition]]
==References==
<references>
<ref name="higham1986">
{{cite journal
| last=Higham | first=Nicholas J.
| title=Computing the polar decomposition with applications
| journal=SIAM J. Sci. Stat. Comput. | volume=7 | number=4
| year=1986 | issn=0196-5204| pages=1160–1174
| doi=10.1137/0907079
| url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.137.7354
| publisher=Society for Industrial and Applied Mathematics
| address=Philadelphia, PA, USA
}}
</ref>
<ref name="byers2008">
{{cite journal
| last=Byers | first=Ralph | coauthors=Hongguo Xu
| title=A New Scaling for Newton's Iteration for the Polar Decomposition and its Backward Stability
| journal=SIAM J. Matrix Anal. Appl.
| volume=30
| number = 2
| year = 2008
| issn = 0895-4798
| pages = 822–843
| doi = 10.1137/070699895
| publisher = Society for Industrial and Applied Mathematics
| address = Philadelphia, PA, USA
}}
</ref>
</references>

== Literature ==
*Conway, J.B.: A Course in Functional Analysis. Graduate Texts in Mathematics. New York: Springer 1990
*Douglas, R.G.: On Majorization, Factorization, and Range Inclusion of Operators on Hilbert Space. Proc. Amer. Math. Soc. '''17''', 413-415 (1966)
* {{citation|first=Sigurdur|last= Helgason|title=Differential geometry, Lie groups, and symmetric spaces|year=1978|publisher=Academic Press|isbn= 0-8218-2848-7}} 

[[Category:Lie groups]]
[[Category:Operator theory]]
[[Category:Matrix theory]]
[[Category:Matrix decompositions]]