Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Generalized inverse
Summary
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 .
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.