Generalized inverse

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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 matrix is a generalized inverse of a matrix if [1][2][3] A generalized inverse exists for an arbitrary matrix, and when a matrix has a regular inverse, this inverse is its unique generalized inverse.[1]

Motivation

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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  

 [4]

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  [1][2][3] The matrix   has been termed a regular inverse of   by some authors.[5]

Types

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Important types of generalized inverse include:

  • One-sided inverse (right inverse or left inverse)
    • Right inverse: If the matrix   has dimensions   and  , then there exists an   matrix   called the right inverse of   such that  , where   is the   identity matrix.
    • Left inverse: If the matrix   has dimensions   and  , then there exists an   matrix   called the left inverse of   such that  , where   is the   identity matrix.[6]
  • Bott–Duffin inverse
  • Drazin inverse
  • Moore–Penrose inverse

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

  1.  
  2.  
  3.  
  4.  

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.[2][7][8][9][10][11] 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.[1]

When   is non-singular, any generalized inverse   and is therefore unique. For a singular  , some generalised inverses, such as the Drazin inverse and the Moore–Penrose inverse, are unique, while others are not necessarily uniquely defined.

Examples

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Reflexive generalized inverse

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Let

 

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  .

One-sided inverse

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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)

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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).

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  .

Construction

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The following characterizations are easy to verify:

  • A right inverse of a non-square matrix   is given by  , provided   has full row rank.[6]
  • A left inverse of a non-square matrix   is given by  , provided   has full column rank.[6]
  • 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, let where   is the non-singular submatrix of  . Then, is a generalized inverse of   if and only if  .

Uses

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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.[12]

Generalized inverses of matrices

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The generalized inverses of matrices can be characterized as follows. Let  , and

 

be its singular-value decomposition. Then for any generalized inverse  , there exist[1] matrices  ,  , and   such that

 

Conversely, any choice of  ,  , and   for matrix of this form is a generalized inverse of  .[1] 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

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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,[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.

See also

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Citations

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  1. ^ a b c d e f Ben-Israel & Greville 2003, pp. 2, 7
  2. ^ a b c Nakamura 1991, pp. 41–42
  3. ^ a b Rao & Mitra 1971, pp. vii, 20
  4. ^ Rao & Mitra 1971, p. 24
  5. ^ Rao & Mitra 1971, pp. 19–20
  6. ^ a b c Rao & Mitra 1971, p. 19
  7. ^ Rao & Mitra 1971, pp. 20, 28, 50–51
  8. ^ Ben-Israel & Greville 2003, p. 7
  9. ^ Campbell & Meyer 1991, p. 10
  10. ^ James 1978, p. 114
  11. ^ Nakamura 1991, p. 42
  12. ^ James 1978, pp. 109–110
  13. ^ Uhlmann 2018

Sources

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Textbook

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  • 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.
  • 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.

Publication

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