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. 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 .
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. 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
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.
The Penrose conditions define different generalized inverses for and
Where indicates 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 . If it satisfies all four conditions, then it is a pseudoinverse of , denoted by . A pseudoinverse is sometimes called the Moore–Penrose inverse, after the pioneering works by E. H. Moore and Roger Penrose. 
When is non-singular, any generalized inverse and is unique. Otherwise, there are an infinite number of -inverses for a given with less than 4 elements. However, the pseudoinverse is unique.
There are other kinds of generalized inverse:
- One-sided inverse (right inverse or left inverse)
Reflexive generalized inverse
Since , is singular and has no regular inverse. However, and satisfy Penrose conditions (1) and (2), but not (3) or (4). Hence, is a reflexive generalized inverse of .
Since is not square, has no regular inverse. However, is a right inverse of . The matrix has no left inverse.
Inverse of other semigroups (or rings)
The generalized inverses of the element 3 in the ring are 3, 7, and 11, since in the ring :
The generalized inverses of the element 4 in the ring are 1, 4, 7, and 10, since in the ring :
If an element a in a semigroup (or ring) has an inverse, the inverse must be the only generalized inverse of this element, like the elements 1, 5, 7, and 11 in the ring .
In the ring , any element is a generalized inverse of 0, however, 2 has no generalized inverse, since there is no b in such that .
The following characterizations are easy to verify:
- A right inverse of a non-square matrix is given by , provided has full row rank.
- A left inverse of a non-square matrix is given by , provided has full column rank.
- 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, letwhere is the non-singular submatrix of . Then,is a generalized inverse of .
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
with vector of unknowns and vector of constants, all solutions are given by
parametric on the arbitrary vector , where is any generalized inverse of . Solutions exist if and only if is a solution, that is, if and only if . If A has full column rank, the bracketed expression in this equation is the zero matrix and so the solution is unique.
The Penrose conditions are used to classify and study different generalized inverses. As noted above, the Penrose conditions are:
In contrast to above, here the numbering of the properties is relevant; this numbering is used in the literature. An -inverse of , where , is a generalized inverse of which satisfies the Penrose conditions listed in . For example, a reflexive inverse is a -inverse, a right-inverse is a -inverse, a left-inverse is a -inverse, and a pseudoinverse is a -inverse. Much research has been devoted to the study between these different classes of generalized inverses; many such results can be found from reference. An example of such a result is the following:
- for any -inverse and -inverse . In particular, for any -inverse .
The section on generalized inverses of matrices provides explicit characterizations for the different classes.
Generalized inverses of matrices
The generalized inverses of matrices can be characterized as follows. Let , and
Conversely, any choice of , , and for matrix of this form is a generalized inverse of . The -inverses are exactly those for which , the -inverses are exactly those for which , and the -inverses are exactly those for which . In particular, the pseudoinverse is given by :
Transformation consistency properties
In practical applications it is necessary to identify the class of matrix transformations that must be preserved by a generalized inverse. For example, the Moore–Penrose inverse, satisfies the following definition of consistency with respect to transformations involving unitary matrices U and V:
The Drazin inverse, satisfies the following definition of consistency with respect to similarity transformations involving a nonsingular matrix S:
The unit-consistent (UC) inverse, satisfies the following definition of consistency with respect to transformations involving nonsingular diagonal matrices D and E:
The fact that the Moore–Penrose inverse provides consistency with respect to rotations (which are orthonormal transformations) explains its widespread use in physics and other applications in which Euclidean distances must be preserved. The UC inverse, by contrast, is applicable when system behavior is expected to be invariant with respect to the choice of units on different state variables, e.g., miles versus kilometers.
- Ben-Israel & Greville 2003, pp. 2, 7
- Nakamura 1991, pp. 41–42
- Rao & Mitra 1971, pp. vii, 20
- Rao & Mitra 1971, p. 24
- Rao & Mitra 1971, pp. 19–20
- Ben-Israel & Greville 2003, p. 7
- Campbell & Meyer 1991, p. 9
- Rao & Mitra 1971, pp. 20, 28, 51
- Campbell & Meyer 1991, p. 10
- James 1978, p. 114
- Nakamura 1991, p. 42
- Rao & Mitra 1971, p. 50–51
- James 1978, pp. 113–114
- Rao & Mitra 1971, p. 19
- James 1978, pp. 109–110
- Uhlmann 2018
- Ben-Israel, Adi; Greville, Thomas Nall Eden (2003). Generalized Inverses: Theory and Applications (2nd ed.). New York, NY: Springer. doi:10.1007/b97366. ISBN 978-0-387-00293-4.
- Campbell, Stephen L.; Meyer, Carl D. (1991). Generalized Inverses of Linear Transformations. Dover. ISBN 978-0-486-66693-8.
- Horn, Roger Alan; Johnson, Charles Royal (1985). Matrix Analysis. Cambridge University Press. ISBN 978-0-521-38632-6. CS1 maint: discouraged parameter (link)
- Nakamura, Yoshihiko (1991). Advanced Robotics: Redundancy and Optimization. Addison-Wesley. ISBN 978-0201151985.
- Rao, C. Radhakrishna; Mitra, Sujit Kumar (1971). Generalized Inverse of Matrices and its Applications. New York: John Wiley & Sons. pp. 240. ISBN 978-0-471-70821-6.
- James, M. (June 1978). "The generalised inverse". The Mathematical Gazette. 62 (420): 109–114. doi:10.2307/3617665. JSTOR 3617665.
- Uhlmann, Jeffrey K. (2018). "A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations" (PDF). SIAM Journal on Matrix Analysis and Applications. 239 (2): 781–800. doi:10.1137/17M113890X.
- Zheng, Bing; Bapat, Ravindra (2004). "Generalized inverse A(2)T,S and a rank equation". Applied Mathematics and Computation. 155 (2): 407–415. doi:10.1016/S0096-3003(03)00786-0.