In mathematics, a generalized inverse or pseudoinverse of a matrixA 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.
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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 0-387-00293-6Zbl1026.15004
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