Matrix decomposition: Difference between revisions

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=== Mostow's decomposition ===

{{main|Mostow 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|last=Nielsen|first=Frank|last2=Bhatia|first2=Rajendra|publisher=Springer|year=2012|isbn=9783642302329|___location=|pages=224|language=en|doi=10.1007/978-3-642-30232-9|arxiv = 1007.4402}}</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 ''U₂'' is unitary, ''M₂'' is real anti-symmetric and ''S₂'' is real symmetric.<ref name=":0" />

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== References ==

*{{cite journal|last1=Choudhury|first1=Dipa|last2=Horn|first2=Roger A.|title=A Complex Orthogonal-Symmetric Analog of the Polar Decomposition|journal=SIAM Journal on Algebraic and Discrete Methods|date=April 1987|volume=8|issue=2|pages=219–225|doi=10.1137/0608019}}

*{{citation|first=I.|last=Fredholm|title=Sur une classe d'´equations fonctionnelles|journal=Acta Mathematica|volume=27|pages=365–390|year=1903|language=Frenchfr|authorlinkauthor-link=Ivar Fredholm|doi=10.1007/bf02421317|doi-access=free}}

*{{citation|first=D.|last=Hilbert|title=Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen|journal=Nachr. Königl. Ges. Gött|volume=1904|pages=49–91|year=1904|language=Germande|authorlinkauthor-link=David Hilbert}}

*{{cite journal|last1=Horn|first1=Roger A.|last2=Merino|first2=Dennis I.|title=Contragredient equivalence: A canonical form and some applications|journal=Linear Algebra and Its Applications|date=January 1995|volume=214|pages=43–92|doi=10.1016/0024-3795(93)00056-6|doi-access=free}}

*{{Citation|last1=Meyer|first1=C. D.|title=Matrix Analysis and Applied Linear Algebra|url=http://www.matrixanalysis.com/|publisher=[[Society for Industrial and Applied Mathematics|SIAM]]|isbn=978-0-89871-454-8|year=2000}}

*{{citation|first=E.|last=Schmidt|title=Zur Theorie der linearen und nichtlinearen Integralgleichungen. I Teil. Entwicklung willkürlichen Funktionen nach System vorgeschriebener|journal=Mathematische Annalen|volume=63|issue=4|pages=433–476|year=1907|language=Germande|authorlinkauthor-link=Erhard Schmidt|doi=10.1007/bf01449770|url=https://zenodo.org/record/1428258}}

*{{Cite book|last=Simon|first=C.|last2=Blume|first2=L.|year=1994|title=Mathematics for Economists|publisher= Norton|isbn=978-0-393-95733-4|ref=harv}}

*{{citation|last=Stewart|first=G. W.|year=2011|title=Fredholm, Hilbert, Schmidt: three fundamental papers on integral equations|url=http://www.cs.umd.edu/~stewart/FHS.pdf|access-date=2015-01-06}}

*{{citation|last=Townsend|first=A.|last2=Trefethen|first2=L. N.|year=2015|title=Continuous analogues of matrix factorizations|journal=[[Proceedings of the Royal Society|Proc. R. Soc. A]]|volume=471|issue=2173|pages=20140585|doi=10.1098/rspa.2014.0585|pmid=25568618|pmc=4277194|bibcode=2014RSPSA.47140585T}}