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
is a generalized inverse of a matrix
such that Hence we can define the generalized inverse as follows: Given an
has been termed a regular inverse of
[5] Important types of generalized inverse include: Some generalized inverses are defined and classified based on the Penrose conditions: where
denotes conjugate transpose.
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
and also known as the Moore–Penrose inverse, after the pioneering works by E. H. Moore and Roger Penrose.
as an inverse that satisfies the subset
is non-singular, any generalized inverse
is singular and has no regular inverse.
satisfy Penrose conditions (1) and (2), but not (3) or (4).
is a reflexive generalized inverse of
The generalized inverses of the element 3 in the ring
: The generalized inverses of the element 4 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
The following characterizations are easy to verify: 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 constants, all solutions are given by parametric on the arbitrary vector
If A has full column rank, the bracketed expression in this equation is the zero matrix and so the solution is unique.
[12] The generalized inverses of matrices can be characterized as follows.
for matrix of this form is a generalized inverse of
In practical applications it is necessary to identify the class of matrix transformations that must be preserved by a generalized 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,[13]
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.