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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. 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 .

Formally, given a matrix and a matrix , is a generalized inverse of if it satisfies the condition .[1][2][3]

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.[4]

Contents

MotivationEdit

Consider the linear system

 

where   is an   matrix and  , the column space of  . If   is nonsingular (which implies  ) 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  ,

 [5]

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 or g-inverse as follows: Given an   matrix  , an   matrix   is said to be a generalized inverse of   if  [6][7][8] The matrix   has been termed a regular inverse of   by some authors.[9]

TypesEdit

The Penrose conditions define different generalized inverses for   and  

  1.  
  2.  
  3.  
  4.  

where   indicates 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  .[10][11][12][13] A pseudoinverse is sometimes called the Moore–Penrose inverse, after the pioneering works by E. H. Moore and Roger Penrose.[14][15][16][17][18]

When   is non-singular, any generalized inverse   and is unique, but in all other cases, there are an infinite number of matrices that satisfy condition (1). However, the Moore–Penrose inverse is unique.[19]

There are other kinds of generalized inverse:

  • 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.[20]

ExamplesEdit

Reflexive generalized inverseEdit

Let

 

Since  ,   is singular and has no regular inverse. However,   and   satisfy conditions (1) and (2), but not (3) or (4). Hence,   is a reflexive generalized inverse of  .

One-sided inverseEdit

Let

 

Since   is not square,   has no regular inverse. However,   is a right inverse of  . The matrix   has no left inverse.

ConstructionEdit

The following characterizations are easy to verify:

  1. A right inverse of a non-square matrix   is given by  , provided A has full row rank.[21]
  2. A left inverse of a non-square matrix   is given by  , provided A has full column rank.[22]
  3. If   is a rank factorization, then   is a g-inverse of  , where   is a right inverse of   and   is left inverse of  .
  4. If   for any non-singular matrices   and  , then   is a generalized inverse of   for arbitrary   and  .
  5. Let   be of rank  . Without loss of generality, let
     

    where   is the non-singular submatrix of  . Then,

     
    is a generalized inverse of  .
  6. Let   have singular-value decomposition   (where   is the conjugate transpose of  ). Then the pseudoinverse of   is
     
    where the diagonal matrix Σ+ is the pseudoinverse of Σ, which is formed by replacing every non-zero diagonal entry by its reciprocal and transposing the resulting matrix.[23]

UsesEdit

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.[24]

Transformation consistency propertiesEdit

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,[25]  , 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 alsoEdit

NotesEdit

  1. ^ Ben-Israel & Greville (2003, pp. 2,7)
  2. ^ Nakamura (1991, pp. 41–42)
  3. ^ Rao & Mitra (1971, pp. vii,20)
  4. ^ Ben-Israel & Greville (2003, pp. 2,7)
  5. ^ Rao & Mitra (1971, p. 24)
  6. ^ Ben-Israel & Greville (2003, pp. 2,7)
  7. ^ Nakamura (1991, pp. 41–42)
  8. ^ Rao & Mitra (1971, pp. vii,20)
  9. ^ Rao & Mitra (1971, pp. 19–20)
  10. ^ Ben-Israel & Greville (2003, p. 7)
  11. ^ Campbell & Meyer (1991, p. 9)
  12. ^ Nakamura (1991, pp. 41–42)
  13. ^ Rao & Mitra (1971, pp. 20,28,51)
  14. ^ Ben-Israel & Greville (2003, p. 7)
  15. ^ Campbell & Meyer (1991, p. 10)
  16. ^ James (1978, p. 114)
  17. ^ Nakamura (1991, p. 42)
  18. ^ Rao & Mitra (1971, p. 50–51)
  19. ^ James (1978, pp. 113–114)
  20. ^ Rao & Mitra (1971, p. 19)
  21. ^ Rao & Mitra (1971, p. 19)
  22. ^ Rao & Mitra (1971, p. 19)
  23. ^ Horn & Johnson (1985, pp. 421)
  24. ^ James (1978, pp. 109–110)
  25. ^ Uhlmann, J.K. (2018), A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations, SIAM Journal on Matrix Analysis, 239:2, pp. 781–800

ReferencesEdit