Generalized inverse

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 formula_1.
A matrix formula_2 is a generalized inverse of a matrix formula_3 if formula_4
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.
Motivation.
Consider the linear system
where formula_1 is an formula_7 matrix and formula_8 the column space of formula_1. If formula_1 is nonsingular (which implies formula_11) then formula_12 will be the solution of the system. Note that, if formula_1 is nonsingular, then
Now suppose formula_1 is rectangular (formula_16), or square and singular. Then we need a right candidate formula_17 of order formula_18 such that for all formula_8
That is, formula_21 is a solution of the linear system formula_5. 
Equivalently, we need a matrix formula_17 of order formula_24 such that
Hence we can define the generalized inverse as follows: Given an formula_18 matrix formula_1, an formula_7 matrix formula_17 is said to be a generalized inverse of formula_1 if formula_25 The matrix formula_32 has been termed a regular inverse of formula_1 by some authors.
Types.
The Penrose conditions define different generalized inverses for formula_34 and formula_35
Where formula_40 indicates conjugate transpose. If formula_41 satisfies the first condition, then it is a generalized inverse of formula_1. If it satisfies the first two conditions, then it is a reflexive generalized inverse of formula_1. If it satisfies all four conditions, then it is a pseudoinverse of formula_1, denoted by formula_45. A pseudoinverse is sometimes called the Moore–Penrose inverse, after the pioneering works by E. H. Moore and Roger Penrose. 
When formula_1 is non-singular, any generalized inverse formula_47 and is unique. Otherwise, there are an infinite number of formula_48-inverses for a given formula_48 with less than 4 elements. However, the pseudoinverse is unique.
There are other kinds of generalized inverse:
Examples.
Reflexive generalized inverse.
Let
Since formula_69, formula_70 is singular and has no regular inverse. However, formula_70 and formula_72 satisfy Penrose conditions (1) and (2), but not (3) or (4). Hence, formula_72 is a reflexive generalized inverse of formula_70.
One-sided inverse.
Let
Since formula_70 is not square, formula_70 has no regular inverse. However, formula_78 is a right inverse of formula_70. The matrix formula_70 has no left inverse.
Inverse of other semigroups (or rings).
The element "b" is a generalized inverse of an element "a" if and only if formula_81, in any semigroup (or ring, since the multiplication function in any ring is a semigroup).
The generalized inverses of the element 3 in the ring formula_82 are 3, 7, and 11, since in the ring formula_82:
The generalized inverses of the element 4 in the ring formula_82 are 1, 4, 7, and 10, since in the ring formula_82:
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 formula_82.
In the ring formula_82, any element is a generalized inverse of 0, however, 2 has no generalized inverse, since there is no "b" in formula_82 such that formula_96.
Construction.
The following characterizations are easy to verify:
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
with vector formula_125 of unknowns and vector formula_126 of constants, all solutions are given by 
parametric on the arbitrary vector formula_128, where formula_41 is any generalized inverse of formula_1. Solutions exist if and only if formula_131 is a solution, that is, if and only if formula_132. If "A" has full column rank, the bracketed expression in this equation is the zero matrix and so the solution is unique.
Penrose conditions.
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 formula_48-inverse of formula_1, where formula_139, is a generalized inverse of formula_1 which satisfies the Penrose conditions listed in formula_48. For example, a reflexive inverse is a formula_142-inverse, a right-inverse is a formula_143-inverse, a left-inverse is a formula_144-inverse, and a pseudoinverse is a formula_145-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:
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 formula_3, and 
formula_155 
be its singular-value decomposition. Then for any generalized inverse formula_156, there exist matrices formula_157, formula_158, and formula_159 such that
formula_160
Conversely, any choice of formula_157, formula_158, and formula_159 for matrix of this form is a generalized inverse of formula_1. The formula_165-inverses are exactly those for which formula_166, the formula_149-inverses are exactly those for which formula_168, and the formula_147-inverses are exactly those for which formula_170. In particular, the pseudoinverse is given by formula_171:
formula_172
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, formula_173 satisfies the following definition of consistency with respect to transformations involving unitary matrices "U" and "V":
The Drazin inverse, formula_175 satisfies the following definition of consistency with respect to similarity transformations involving a nonsingular matrix "S":
The unit-consistent (UC) inverse, formula_177 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.