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 matrix is a generalized inverse of a matrix if 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 is an matrix and the column space of . If is nonsingular (which implies ) then will be the solution of the system. Note that, if is nonsingular, then Now suppose is rectangular (), or square and singular. Then we need a right candidate of order such that for all That is, is a solution of the linear system . Equivalently, we need a matrix of order such that Hence we can define the generalized inverse as follows: Given an matrix , an matrix is said to be a generalized inverse of if The matrix has been termed a regular inverse of by some authors. Important types of generalized inverse include: One-sided inverse (right inverse or left inverse) Right inverse: If the matrix has dimensions and , then there exists an matrix called the right inverse of such that , where is the identity matrix. Left inverse: If the matrix has dimensions and , then there exists an matrix called the left inverse of such that , where is the identity matrix. Bott–Duffin inverse Drazin inverse Moore–Penrose inverse Some generalized inverses are defined and classified based on the Penrose conditions: where denotes conjugate transpose. If satisfies the first condition, then it is a generalized inverse of . If it satisfies the first two conditions, then it is a reflexive generalized inverse of .