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 and a matrix , is a generalized inverse of if it satisfies the condition .
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 ones. 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
Suppose the matrix is singular or then we need a right candidate of order such that
That is is a solution of the linear system . Equivalently, of order such that
Hence we can define the generalized inverse as follows: Given a matrix , a matrix is said to be generalized inverse of if
Construction of generalized inverse
The following characterizations are easy to verify.
- If is a rank factorization, then is a g-inverse of where is a right inverse of and is left inverse of .
- If for any non-singular matrices and , then is a generalized inverse of for arbitrary and .
- Let be of rank . Without loss of generality, let
- where is the non-singular submatrix of . Then,
- is a g-inverse of .
Types of generalized inverses
The Penrose conditions are used to define different generalized inverses: for and
If satisfies condition (1.), it is a generalized inverse of , if it satisfies conditions (1.) and (2.) then it is a generalized reflexive inverse of , and if it satisfies all 4 conditions, then it is a Moore–Penrose pseudoinverse of .
Other various kinds of generalized inverses include
- One-sided inverse (left inverse or right inverse) If the matrix A has dimensions and is full rank then use the left inverse if and the right inverse if
- Left inverse is given by , i.e. where is the identity matrix.
- Right inverse is given by , i.e. where is the identity matrix.
- Drazin inverse
- Bott–Duffin inverse
- Moore–Penrose pseudoinverse
with vector of unknowns and vector b of constants, all solutions are given by
parametric on the arbitrary vector w, where is any generalized inverse of Solutions exist if and only if is a solution – that is, if and only if
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- 15A09 Matrix inversion, generalized inverses in Mathematics Subject Classification, MathSciNet search
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