Generalized inverse

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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]

Motivation[edit]

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]

Types[edit]

The Penrose conditions define different generalized inverses for and

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]

Examples[edit]

Reflexive generalized inverse[edit]

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 inverse[edit]

Let

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

Construction[edit]

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]

Uses[edit]

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 Properties[edit]

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 also[edit]

Notes[edit]

  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 

References[edit]