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en>Discospinster: Reverted edits by 189.135.148.97 (talk) to last version by Discospinster

2014-08-25T23:08:45Z

Reverted edits by 189.135.148.97 (talk) to last version by Discospinster

← Older revision		Revision as of 01:08, 26 August 2014
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	~~{{redirect\|Pseudoinverse\|the Moore-Penrose pseudoinverse, sometimes referred to as "the pseudoinverse"\|Moore–Penrose pseudoinverse}}~~		I'm Luther but I by no means really [http://En.Wikipedia.org/wiki/Warranty favored] that name. Kansas extended car [http://Autos.Aol.com/article/whats-in-car-warranty/ warranty auto] warranty is where her home is but she requirements to move because of extended car [http://www.amicididonbosco.org/ActivityFeed/MyProfile/tabid/61/userId/334/language/it-IT/Default.aspx auto warranty] her family. His day occupation is a cashier and his salary has been truly fulfilling. The preferred hobby for my children and me is playing car warranty crochet and now I'm attempting to make money with it.<br><br>Here is my web page :: extended [http://hs-s.com/dayz/home/index.php?mod=users&action=view&id=8983 car warranty] ([http://Childsearchandrescue.org/node/601730 click through the next site])
	~~In [[mathematics]], a~~ '~~''generalized inverse''' of a~~ [[matrix (mathematics)\|matrix]] ''A'' is a matrix that has some properties of the [[inverse matrix]] of ''A'' but not necessarily all of them. Formally, given a matrix <math>A \in \mathbb{R}^{n\times m}</~~math> and a matrix <math>X \in \mathbb{R}^{m\times n}<~~/~~math>, <math>X<~~/~~math> is a generalized inverse of <math>A<~~/~~math> if it satisfies the condition <math> A X A = A</math>.~~

	~~The purpose of constructing a generalized inverse is to obtain a matrix~~ that ~~can serve as the inverse in some sense for a wider class of matrices than invertible ones~~. ~~Typically, the generalized inverse exists for an arbitrary matrix, and when a matrix has an inverse, then its inverse and the generalized inverse are the same. Some~~ ~~generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a~~ [~~[semigroup]].~~

	~~== Types of generalized inverses ==~~
	~~The Penrose conditions are used to define different generalized inverses~~: ~~for <math>A \in \mathbb{R}^{n\times m}<~~/~~math> and <math>X \in \mathbb{R}^{m\times n}<~~/~~math>,~~
	{\|
	~~\| 1~~.~~) \|\| <math>AXA = A<~~/~~math>~~
	\|-
	~~\| 2.) \|\| <math>XAX = X</math>~~
	\|-
	~~\| 3.) \|\| <math>(AX)^{T} = AX</math>~~
	\|-
	~~\| 4.) \|\| <math>(XA)^{T} = XA<~~/~~math> .~~
	\|}

	~~If <math>X</math> satisfies condition (1.), it~~ is ~~a generalized inverse~~ of ~~<math>A<~~/~~math>, if it satisfies conditions (1~~.~~) and (2~~.~~) then it is a generalized reflexive inverse of <math>A<~~/~~math> , and if it satisfies all 4 conditions, then it is a [[pseudoinverse]] of <math>A<~~/~~math>.~~

	~~Other various kinds of generalized inverses include~~
	* [[One-sided inverse]] (left inverse or right inverse) If the matrix ''A'' has dimensions <math>m \times n</~~math> and is [[full rank]] then use the left inverse if <math>m > n<~~/~~math> and the right inverse if <math>m < n<~~/~~math>~~
	** Left inverse is given by <math>A_{\mathrm{left}}^{-1} = \left(A^T A\right)^{-1} A^T</~~math>, i.e. <math>A_{\mathrm{left}}^{-1} A = I_n<~~/~~math> where <math>I_n<~~/~~math> is the <math>n \times n<~~/~~math> [[identity matrix]~~].
	** Right inverse is ~~given by <math>A_{\mathrm{right}}^{-1} = A^T \left(A A^T\right)^{-1}</math>, i~~.e. <~~math~~>~~A A_{\mathrm{right}}^{-1} = I_m~~<~~/math> where <math>I_m</math~~> is ~~the <math>m \times m</math> identity matrix.~~
	* [~~[Drazin inverse]]~~
	* [[Bott–Duffin inverse]]
	* [[Moore–Penrose pseudoinverse]]

	~~== See also ==~~
	* [[Inverse element]]

	~~== References ==~~
	* {{cite book\| authors= Yoshihiko Nakamura \| title= * Advanced Robotics: Redundancy and Optimization\|
	~~publisher=Addison-Wesley \|year= 1991 \|ISBN =0201151987}}~~
	* {{cite journal\| last1=Zheng\| first1=B\| last2=Bapat\|first2= R. B.\| title=Generalized inverse A(2)T,S and a rank equation\| journal=Applied Mathematics and Computation\| volume=155\| pages=407–415\| year=2004\| doi=10.1016/S0096-3003(03)00786-0}}
	* {{cite book\| authors= S. L. Campbell and C. D. Meyer \| title= Generalized Inverses of Linear Transformations\|
	~~publisher=Dover \|year=1991 \|ISBN =978-0-486-66693-8}}~~
	* {{cite book\| authors= Adi Ben-Israel and Thomas N.E. Greville \|title=Generalized inverses. Theory and applications\|
	~~edition= 2nd \|location= New York, NY \| publisher = Springer \|year=2003\| ISBN = 0-387-00293-6\| url =~~ http://~~link.springer~~.com/~~book~~/~~10.1007~~/~~b97366/page/1}}~~
	* {{cite book\| authors=C. ~~R. Rao and C. Radhakrishna Rao and Sujit Kumar Mitra\| title~~ = ~~Generalized Inverse of Matrices and its Applications \| publisher~~= ~~John Wiley~~ & ~~Sons \|location = New York \|year= 1971 \| pages=240 \|ISBN =0-471-70821-6}}~~

	=~~= External links ==~~
	* [http://www.ams.org/msc/15-xx.html 15A09] Matrix inversion, generalized inverses in [[Mathematics Subject Classification]], [[MathSciNet]] [http://~~www.ams~~.org/~~mathscinet~~/~~search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=15A09&co6=AND&pg7=ALLF&s7=&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All search~~]

	~~[[Category:Matrices]]~~
	~~[[Category:Mathematical terminology]]~~


	~~{{Linear-algebra-stub}}~~

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Bot: Migrating 12 interwiki links, now provided by Wikidata on d:q923233 (Report Errors)

← Older revision		Revision as of 21:30, 25 February 2013
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	The ~~writer~~ is ~~known~~ as ~~Wilber Pegues~~. ~~Distributing manufacturing~~ is ~~exactly where her primary earnings arrives from~~. ~~To play domino~~ is ~~something I really appreciate performing~~. I'~~ve always cherished residing in Mississippi~~.<br><br>~~Here~~ is ~~my weblog~~ ... [http://~~checkmates~~.co.za/~~index~~.~~php?do~~=/~~profile~~-~~56347~~/~~info~~/ ~~real psychic readings~~]		{{redirect\|Pseudoinverse\|the Moore-Penrose pseudoinverse, sometimes referred to as "the pseudoinverse"\|Moore–Penrose pseudoinverse}}
			In [[mathematics]], a '''generalized inverse''' of a [[matrix (mathematics)\|matrix]] ''A'' is a matrix that has some properties of the [[inverse matrix]] of ''A'' but not necessarily all of them. Formally, given a matrix <math>A \in \mathbb{R}^{n\times m}</math> and a matrix <math>X \in \mathbb{R}^{m\times n}</math>, <math>X</math> is a generalized inverse of <math>A</math> if it satisfies the condition <math> A X A = A</math>.

			The purpose of constructing a generalized inverse is to obtain a matrix that can serve as the inverse in some sense for a wider class of matrices than invertible ones. Typically, the generalized inverse exists for an arbitrary matrix, and when a matrix has an inverse, then its inverse and the generalized inverse are the same. Some generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a [[semigroup]].

			== Types of generalized inverses ==
			The Penrose conditions are used to define different generalized inverses: for <math>A \in \mathbb{R}^{n\times m}</math> and <math>X \in \mathbb{R}^{m\times n}</math>,
			{\|
			\| 1.) \|\| <math>AXA = A</math>
			\|-
			\| 2.) \|\| <math>XAX = X</math>
			\|-
			\| 3.) \|\| <math>(AX)^{T} = AX</math>
			\|-
			\| 4.) \|\| <math>(XA)^{T} = XA</math> .
			\|}

			If <math>X</math> satisfies condition (1.), it is a generalized inverse of <math>A</math>, if it satisfies conditions (1.) and (2.) then it is a generalized reflexive inverse of <math>A</math> , and if it satisfies all 4 conditions, then it is a [[pseudoinverse]] of <math>A</math>.

			Other various kinds of generalized inverses include
			* [[One-sided inverse]] (left inverse or right inverse) If the matrix ''A'' has dimensions <math>m \times n</math> and is [[full rank]] then use the left inverse if <math>m > n</math> and the right inverse if <math>m < n</math>
			** Left inverse is given by <math>A_{\mathrm{left}}^{-1} = \left(A^T A\right)^{-1} A^T</math>, i.e. <math>A_{\mathrm{left}}^{-1} A = I_n</math> where <math>I_n</math> is the <math>n \times n</math> [[identity matrix]].
			** Right inverse is given by <math>A_{\mathrm{right}}^{-1} = A^T \left(A A^T\right)^{-1}</math>, i.e. <math>A A_{\mathrm{right}}^{-1} = I_m</math> where <math>I_m</math> is the <math>m \times m</math> identity matrix.
			* [[Drazin inverse]]
			* [[Bott–Duffin inverse]]
			* [[Moore–Penrose pseudoinverse]]

			== See also ==
			* [[Inverse element]]

			== References ==
			* {{cite book\| authors= Yoshihiko Nakamura \| title= * Advanced Robotics: Redundancy and Optimization\|
			publisher=Addison-Wesley \|year= 1991 \|ISBN =0201151987}}
			* {{cite journal\| last1=Zheng\| first1=B\| last2=Bapat\|first2= R. B.\| title=Generalized inverse A(2)T,S and a rank equation\| journal=Applied Mathematics and Computation\| volume=155\| pages=407–415\| year=2004\| doi=10.1016/S0096-3003(03)00786-0}}
			* {{cite book\| authors= S. L. Campbell and C. D. Meyer \| title= Generalized Inverses of Linear Transformations\|
			publisher=Dover \|year=1991 \|ISBN =978-0-486-66693-8}}
			* {{cite book\| authors= Adi Ben-Israel and Thomas N.E. Greville \|title=Generalized inverses. Theory and applications\|
			edition= 2nd \|location= New York, NY \| publisher = Springer \|year=2003\| ISBN = 0-387-00293-6\| url = http://link.springer.com/book/10.1007/b97366/page/1}}
			* {{cite book\| authors=C. R. Rao and C. Radhakrishna Rao and Sujit Kumar Mitra\| title = Generalized Inverse of Matrices and its Applications \| publisher= John Wiley & Sons \|location = New York \|year= 1971 \| pages=240 \|ISBN =0-471-70821-6}}

			== External links ==
			* [http://www.ams.org/msc/15-xx.html 15A09] Matrix inversion, generalized inverses in [[Mathematics Subject Classification]], [[MathSciNet]] [http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=15A09&co6=AND&pg7=ALLF&s7=&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All search]

			[[Category:Matrices]]
			[[Category:Mathematical terminology]]


			{{Linear-algebra-stub}}

en>Helpful Pixie Bot: ISBNs (Build KE)

2012-05-10T04:42:42Z

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