Generalized inverse


In mathematics, and in particular, algebra, a generalized inverse of an element x is an element y that has some properties of an inverse element but not necessarily all of them. 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. 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.
A matrix is a generalized inverse of a matrix if 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 is an matrix and the column space of. If and is nonsingular then will be the solution of the system. Note that, if is nonsingular, then
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 problem is how to choose an as the output of for every when the map is not bijective.
  • If is not surjective, then not all 's in its codomain have corresponding 's via. To circumvent it, we just let map those 's to arbitrary values.
  • If is not injective, then some 's correspond to multiple 's via. To circumvent it, we let map every to one of the 's according an algorithm.
  • If is neither surjective nor injective, we combine the above two tricks.

Types

Important types of generalized inverse include:
Some generalized inverses are defined and classified based on the Penrose conditions:
where denotes 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 the pseudoinverse of, which is denoted by and also known as the Moore–Penrose inverse, after the pioneering works by E. H. Moore and Roger Penrose. It is convenient to define an -inverse of as an inverse that satisfies the subset of the Penrose conditions listed above. Relations, such as, can be established between these different classes of -inverses.
When is non-singular, any generalized inverse is equal to and is therefore unique. For a singular, some generalized inverses, such as the Drazin inverse and the Moore–Penrose inverse, are unique, while others are not necessarily uniquely defined.

Examples

Non-reflexive generalized inverse

Let
Obviously, is singular. and satisfy Penrose conditions, but not the other there. Hence, is a non-reflexive generalized inverse of.
The first column of spans, and maps it to, which does not lie in. Additionally, maps to, which lies in. The relationship is summarized in the picture on the right.

Reflexive generalized inverse

Let
Since, is singular and has no regular inverse. However, and satisfy Penrose conditions and, but not or. Hence, is a reflexive generalized inverse of.

One-sided inverse

Let
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 element b is a generalized inverse of an element a if and only if, in any semigroup.
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 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.

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 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.

Generalized inverses of matrices

The generalized inverses of matrices can be characterized as follows. Let, and
be its singular-value decomposition. Then for any generalized inverse, there exist matrices,, and such that
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 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 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.

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