Generalized inverse  In mathematics, and in particular, algebra, a generalized inverse of an element "x" is an element "y" that has some properties of an inverse element but not necessarily all of them. 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 formula_1. Formally, given a matrix formula_2 and a matrix formula_3, formula_4 is a generalized inverse of formula_1 if it satisfies the condition formula_6  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. A generalized inverse exists for an arbitrary matrix, and when a matrix has a regular inverse, this inverse is its unique generalized inverse. Consider the linear system  where formula_1 is an formula_9 matrix and formula_10 the column space of formula_1. If formula_1 is nonsingular (which implies formula_13) then formula_14 will be the solution of the system. Note that, if formula_1 is nonsingular, then  Now suppose formula_1 is rectangular (formula_18), or square and singular. Then we need a right candidate formula_19 of order formula_20 such that for all formula_10  That is, formula_23 is a solution of the linear system formula_7. Equivalently, we need a matrix formula_19 of order formula_26 such that  Hence we can define the generalized inverse or g-inverse as follows: Given an formula_9 matrix formula_1, an formula_20 matrix formula_19 is said to be a generalized inverse of formula_1 if formula_27 The matrix formula_34 has been termed a regular inverse of formula_1 by some authors. The Penrose conditions define different generalized inverses for formula_2 and formula_37   where formula_42 indicates conjugate transpose. If formula_4 satisfies the first condition, then it is a generalized inverse of formula_1. If it satisfies the first two conditions, then it is a reflexive generalized inverse of formula_1. If it satisfies all four conditions, then it is the pseudoinverse of formula_1. A pseudoinverse is sometimes called the Moore–Penrose inverse, after the pioneering works by E. H. Moore and Roger Penrose.