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| {{redirect|Pseudoinverse|the Moore-Penrose pseudoinverse, sometimes referred to as "the pseudoinverse"|Moore–Penrose pseudoinverse}}
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| 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>.
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| 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]].
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| == Types of generalized inverses ==
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| 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>,
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| {|
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| | 1.) || <math>AXA = A</math>
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| |-
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| | 2.) || <math>XAX = X</math>
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| |-
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| | 3.) || <math>(AX)^{T} = AX</math>
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| |-
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| | 4.) || <math>(XA)^{T} = XA</math> .
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| |}
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| 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>.
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| Other various kinds of generalized inverses include
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| * [[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>
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| ** 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]].
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| ** 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.
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| * [[Drazin inverse]]
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| * [[Bott–Duffin inverse]]
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| * [[Moore–Penrose pseudoinverse]]
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| == See also ==
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| * [[Inverse element]]
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| == References ==
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| * {{cite book| authors= Yoshihiko Nakamura | title= * Advanced Robotics: Redundancy and Optimization|
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| publisher=Addison-Wesley |year= 1991 |ISBN =0201151987}}
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| * {{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}}
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| * {{cite book| authors= S. L. Campbell and C. D. Meyer | title= Generalized Inverses of Linear Transformations|
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| publisher=Dover |year=1991 |ISBN =978-0-486-66693-8}}
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| * {{cite book| authors= Adi Ben-Israel and Thomas N.E. Greville |title=Generalized inverses. Theory and applications|
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| 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}}
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| * {{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}}
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| == 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]
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| [[Category:Matrices]]
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| [[Category:Mathematical terminology]]
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| {{Linear-algebra-stub}}
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