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. 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 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

the column space is nonsingular is nonsingular, then

such that

Hence we can define the generalized inverse has been termed a regular inverse by some authors.

is not bijective.

• is not surjective's in its codomain's to arbitrary values.

as the direct sum as follows.

• 's.

• 's).

• respectively, then take the sum.

• is not injective's according an algorithm.

• as the direct sum, its preimage to this point.

• is neither surjective nor injective, we combine the above two tricks.

• The picture on the right is an example.

Types

Important types of generalized inverse include:

One-sided inverse (right inverse or left inverse)

• called the right inverse identity matrix.

• called the left inverse identity matrix.

  Bott–Duffin inverse

• Drazin inverse

• Moore–Penrose inverse

Some generalized inverses are defined and classified based on the Penrose conditions:

denotes conjugate transpose satisfies the first condition, then it is a generalized inverse. If it satisfies the first two conditions, then it is a reflexive generalized inverse. If it satisfies all four conditions, then it is the pseudoinverse 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-inverses.

, 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

.

. The relationship is summarized in the picture on the right.

Reflexive generalized inverse

Let

.

One-sided inverse

Let

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 (or ring, since the multiplication function in any ring is a semigroup).

:

:

If an element a.

any element is a generalized inverse of 0; however 2 has no generalized inverse, since there is no b.

Construction

The following characterizations are easy to verify:

• A right inverse of a non-square matrix has full row rank.

• has full column rank.

• is a rank factorization.

• .

• . Without loss of generality.

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

of constants, all solutions are given by

. 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

, and

be its singular-value decomposition, there exist such that

.:

Transformation consistency properties

satisfies the following definition of consistency with respect to transformations involving unitary matrices U and V:

1. .

satisfies the following definition of consistency with respect to similarity transformations involving a nonsingular matrix S:

1. .

The unit-consistent (UC) inverse, satisfies the following definition of consistency with respect to transformations involving nonsingular diagonal matrices D and E:

1. .

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.

See also

• Block matrix pseudoinverse

• Regular semigroup

Citations

Sources

Textbook

Publication