Rahul Sharma (Editor)

Generalized inverse

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

In mathematics, a generalized inverse of a matrix A is a matrix that has some properties of the inverse matrix of A but not necessarily all of them. Formally, given a matrix A ∈ R n × m and a matrix A g ∈ R m × n , A g is a generalized inverse of A if it satisfies the condition A A g A = A .

Contents

The purpose of constructing a generalized inverse is to obtain a matrix that can serve as the 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 an inverse, then this inverse is its unique generalized inverse. Some generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup.

Motivation for the generalized inverse

Consider the linear system

A x = y

where A is an n × m matrix and y ∈ R ( A ) , the range space of A . If the matrix A is nonsingular, then x = A − 1 y will be the solution of the system. Note that, if a matrix A is nonsingular, then

A A − 1 A = A .

Suppose the matrix A is singular, or n ≠ m . Then we need a right candidate G of order m × n such that for all y ∈ R ( A ) ,

A G y = y .

That is, G y is a solution of the linear system A x = y . Equivalently, we need a matrix G of order m × n such that

A G A = A .

Hence we can define the generalized inverse as follows: Given an n × m matrix A , an m × n matrix G is said to be a generalized inverse of A if A G A = A .

Construction of generalized inverse

The following characterizations are easy to verify:

  1. If A = B C is a rank factorization, then G = C r − B l − is a g-inverse of A , where C r − is a right inverse of C and B l − is left inverse of B .
  2. If A = P [ I r 0 0 0 ] Q for any non-singular matrices P and Q , then G = Q − 1 [ I r U W V ] P − 1 is a generalized inverse of A for arbitrary U , V and W .
  3. Let A be of rank r . Without loss of generality, let

Types of generalized inverses

The Penrose conditions are used to define different generalized inverses for A ∈ R n × m and A g ∈ R m × n :

  1. A A g A = A
  2. A g A A g = A g
  3. ( A A g ) T = A A g
  4. ( A g A ) T = A g A

If A g satisfies the first condition, then it is a generalized inverse of A . If it satisfies the first two conditions, then it is a generalized reflexive inverse of A . If it satisfies all four conditions, then it is a Moore–Penrose pseudoinverse of A .

Other kinds of generalized inverse include:

  • One-sided inverse (left inverse or right inverse): If the matrix A has dimensions n × m and is full rank, then use the left inverse if n > m and the right inverse if n < m .
  • Left inverse is given by A l e f t − 1 = ( A T A ) − 1 A T , i.e., A l e f t − 1 A = I m , where I m is the m × m identity matrix.
  • Right inverse is given by A r i g h t − 1 = A T ( A A T ) − 1 , i.e., A A r i g h t − 1 = I n , where I n is the n × n identity matrix.
  • Drazin inverse
  • Bott–Duffin inverse
  • Moore–Penrose pseudoinverse

    • Uses

    Any generalized inverse can be used to determine whether a system of linear equations has any solutions, and if so to give all of them. If any solutions exist for the n × m linear system

    A x = b ,

    with vector x of unknowns and vector b of constants, all solutions are given by

    x = A g b + [ I − A g A ] w ,

    parametric on the arbitrary vector w , where A g is any generalized inverse of A . Solutions exist if and only if A g b is a solution, that is, if and only if A A g b = b .

    References

    Generalized inverse Wikipedia
    (Text) CC BY-SA