Generalized 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 $A$ .

A matrix $A^{\mathrm {g} }\in \mathbb {R} ^{n\times m}$ is a generalized inverse of a matrix $A\in \mathbb {R} ^{m\times n}$ if $AA^{\mathrm {g} }A=A.$ ^[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]

^ ^a ^b Ben-Israel & Greville 2003, pp. 2, 7
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[:0-1] Ben-Israel & Greville 2003, pp. 2, 7

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