Editing Matrix decomposition (section)

== Other decompositions ==

=== Polar decomposition ===
{{main|Polar decomposition}}
*Applicable to: any square complex matrix ''A''.
*Decomposition: <math>A=UP</math> (right polar decomposition) or <math>A=P'U</math> (left polar decomposition), where ''U'' is a [[unitary matrix]] and ''P'' and ''P''' are [[positive semidefinite matrix|positive semidefinite]] [[Hermitian matrices]].
*Uniqueness: <math>P</math> is always unique and equal to <math>\sqrt{A^*A}</math> (which is always hermitian and positive semidefinite). If <math>A</math> is invertible, then <math>U</math> is unique.
*Comment: Since any Hermitian matrix admits a spectral decomposition with a unitary matrix, <math>P</math> can be written as <math>P=VDV^*</math>. Since <math>P</math> is positive semidefinite, all elements in <math>D</math> are non-negative. Since the product of two unitary matrices is unitary, taking <math>W=UV</math>one can write <math>A=U(VDV^*)=WDV^* </math> which is the singular value decomposition. Hence, the existence of the polar decomposition is equivalent to the existence of the singular value decomposition.

=== Algebraic polar decomposition ===
*Applicable to: square, complex, non-singular matrix ''A''.<ref>{{harvnb|Choudhury|Horn|1987|pp=219–225}}</ref>
*Decomposition: <math>A=QS</math>, where ''Q'' is a complex orthogonal matrix and ''S'' is complex symmetric matrix.
*Uniqueness: If <math>A^\mathsf{T}A</math> has no negative real eigenvalues, then the decomposition is unique.<ref name=":0">{{Cite journal|last=Bhatia|first=Rajendra|date=2013-11-15|title=The bipolar decomposition|journal=Linear Algebra and Its Applications|volume=439|issue=10|pages=3031–3037|doi=10.1016/j.laa.2013.09.006|doi-access=}}</ref>
*Comment: The existence of this decomposition is equivalent to <math>AA^\mathsf{T}</math> being similar to <math>A^\mathsf{T}A</math>.<ref>{{harvnb|Horn|Merino|1995|pp=43–92}}</ref>
*Comment: A variant of this decomposition is <math>A=RC</math>, where ''R'' is a real matrix and ''C'' is a [[circular matrix]].<ref name=":0" />

=== Mostow's decomposition ===
* Applicable to: square, complex, non-singular matrix ''A''.<ref>{{citation|last=Mostow|first= G. D.|title= Some new decomposition theorems for semi-simple groups|series= Mem. Amer. Math. Soc. |year=1955|volume=14|pages= 31–54|url=https://archive.org/details/liealgebrasandli029541mbp|publisher= American Mathematical Society}}</ref><ref>{{Cite book|title=Matrix Information Geometry|last1=Nielsen|first1=Frank|last2=Bhatia|first2=Rajendra|publisher=Springer|year=2012|isbn=9783642302329|pages=224|language=en|doi=10.1007/978-3-642-30232-9|arxiv = 1007.4402|s2cid=118466496 }}</ref>
* Decomposition: <math>A=Ue^{iM}e^{S}</math>, where ''U'' is unitary, ''M'' is real anti-symmetric and ''S'' is real symmetric.
* Comment: The matrix ''A'' can also be decomposed as <math>A=U_2e^{S_2}e^{iM_2}</math>, where ''U''<sub>2</sub> is unitary, ''M''<sub>2</sub> is real anti-symmetric and ''S''<sub>2</sub> is real symmetric.<ref name=":0" />

=== Sinkhorn normal form ===
{{main|Sinkhorn's theorem}}
*Applicable to: square real matrix ''A'' with strictly positive elements.
*Decomposition: <math>A=D_{1}SD_{2}</math>, where ''S'' is [[Doubly stochastic matrix|doubly stochastic]] and ''D''<sub>1</sub> and ''D''<sub>2</sub> are real diagonal matrices with strictly positive elements.

=== Sectoral decomposition ===
*Applicable to: square, complex matrix ''A'' with [[numerical range]] contained in the sector <math>S_\alpha = \left\{r e^{i \theta} \in \mathbb{C} \mid r> 0, |\theta| \le \alpha < \frac{\pi}{2}\right\}</math>.
*Decomposition: <math>A = CZC^*</math>, where ''C'' is an invertible complex matrix and <math>Z = \operatorname{diag}\left(e^{i\theta_1},\ldots,e^{i\theta_n}\right)</math> with all <math>\left|\theta_j\right| \le \alpha </math>.<ref name=Zhang2014>{{cite journal|last1=Zhang|first1=Fuzhen|title=A matrix decomposition and its applications|journal=Linear and Multilinear Algebra|volume=63|issue=10|date=30 June 2014|pages=2033–2042|doi=10.1080/03081087.2014.933219|s2cid=19437967 |url=https://zenodo.org/record/851661}}</ref><ref>{{cite journal|last1=Drury|first1=S.W.|title=Fischer determinantal inequalities and Highamʼs Conjecture|journal=Linear Algebra and Its Applications|date=November 2013|volume=439|issue=10|pages=3129–3133|doi=10.1016/j.laa.2013.08.031|doi-access=free}}</ref>

=== Williamson's normal form ===
* Applicable to: square, [[Positive-definite matrix|positive-definite]] real matrix ''A'' with order 2''n''×2''n''.
* Decomposition: <math>A=S^\mathsf{T}\operatorname{diag}(D,D)S</math>, where <math>S \in \text{Sp}(2n)</math> is a [[symplectic matrix]] and ''D'' is a nonnegative ''n''-by-''n'' diagonal matrix.<ref>{{Cite journal|last1=Idel|first1=Martin|last2=Soto Gaona|first2=Sebastián|last3=Wolf|first3=Michael M.|date=2017-07-15|title=Perturbation bounds for Williamson's symplectic normal form|journal=Linear Algebra and Its Applications|volume=525|pages=45–58|doi=10.1016/j.laa.2017.03.013|arxiv=1609.01338|s2cid=119578994 }}</ref>

===Matrix square root===
{{main|Square root of a matrix}}
* Decomposition: <math>A=BB</math>, not unique in general.
* In the case of positive semidefinite <math>A</math>, there is a unique positive semidefinite <math>B</math> such that <math>A=B^*B=BB</math>.