# Generalized inverse

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Generalized inverse

In mathematics, a generalized inverse or pseudoinverse of a matrix "A" is a matrix that has some properties of the inverse matrix of "A" but not necessarily all of them. The term "the pseudoinverse" commonly means the Moore-Penrose pseudoinverse.

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 various kinds of generalized inverses include
* one-sided inverse, that is left inverse and right inverse
* Drazin inverse
* Group inverse
* Bott–Duffin inverse
* Moore-Penrose pseudoinverse

* Inverse element

References

* Bing Zheng and R. B. Bapat, "Generalized inverse A(2)T,S and a rank equation", Applied Mathematics and Computation 155 (2004) 407-415 [http://dx.doi.org/10.1016/S0096-3003(03)00786-0 DOI 10.1016/S0096-3003(03)00786-0]
* S. L. Campbell and C. D. Meyer, "Generalized Inverses of Linear Transformations", Dover 1991 ISBN 978-0486666938
* Adi Ben-Israel and Thomas N.E. Greville, "Generalized inverses. Theory and applications". 2nd ed. New York, NY: Springer, 2003. ISBN 0-387-00293-6 Zbl [http://www.zentralblatt-math.org/zmath/en/search/?q=an:1026.15004&format=complete 1026.15004]
* C. Radhakrishna Rao and Sujit Kumar Mitra, "Generalized Inverse of Matrices and its Applications", John Wiley & Sons New York, 1971, 240 p., ISBN 0-471-70821-6

* [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]
* [http://mjollnir.com/ Pseudo-Inverse (Not Moore-Penrose)]

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