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. Typically, the generalized inverse exists for an arbitrary matrix, and when a matrix has an inverse, then its inverse and the generalized inverse are the same. Some generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup.
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 x 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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