Algebraic element satisfying some of the criteria of an inverse
"Pseudoinverse" redirects here. For the Moore–Penrose inverse, sometimes referred to as "the pseudoinverse", see Moore–Penrose inverse.
In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element x is an element y that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized inverse of a matrix is to obtain a matrix that can serve as an inverse in some sense for a wider class of matrices than invertible matrices. Generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup. This article describes generalized inverses of a matrix .
A matrix is a generalized inverse of a matrix if [1][2][3] A generalized inverse exists for an arbitrary matrix, and when a matrix has a regular inverse, this inverse is its unique generalized inverse.[1]
^ abBen-Israel & Greville 2003, pp. 2, 7
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