Generalized inverse

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, and in particular, <a href="/facts/Algebra/nMdqczjo">algebra</a>, a generalized inverse (or, g-inverse) of an element x is an element y that has some properties of an <a href="/facts/Inverse_element/CwMygHSj">inverse element</a> 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 <a href="/facts/Invertible_matrix/fPqXk3V8">invertible matrices</a>. Generalized inverses can be defined in any <a href="/facts/Mathematical_structure/6CXReUNv">mathematical structure</a> that involves <a href="/facts/Associative_property/EQX8lV6r">associative</a> multiplication, that is, in a <a href="/facts/Semigroup/rgSdk2kt">semigroup</a>. This article describes generalized inverses of a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> 
 
 
 
 A
 
 
 {\displaystyle A}
 
.
A matrix 
 
 
 
 
 A
 
 
 g
 
 
 
 ∈
 
 
 R
 
 
 n
 ×
 m
 
 
 
 
 {\displaystyle A^{\mathrm {g} }\in \mathbb {R} ^{n\times m}}
 
 is a generalized inverse of a matrix 
 
 
 
 A
 ∈
 
 
 R
 
 
 m
 ×
 n
 
 
 
 
 {\displaystyle A\in \mathbb {R} ^{m\times n}}
 
 if 
 
 
 
 A
 
 A
 
 
 g
 
 
 
 A
 =
 A
 .
 
 
 {\displaystyle AA^{\mathrm {g} }A=A.}
 
 A generalized inverse exists for an arbitrary matrix, and when a matrix has a regular inverse, this inverse is its unique generalized inverse.

Generalized inverse open-in-new

Generalized inverse