Generalized inverse

From Wikipedia, the free encyclopedia

"Pseudoinverse" redirects here. For the Moore-Penrose pseudoinverse, sometimes referred to as "the pseudoinverse", see Moore-Penrose pseudoinverse.

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 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.

1 Types of generalized inverses
2 See also
3 References
4 External links

[edit] 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

[edit] See also

Inverse element

[edit] 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 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 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