Generalized inverse

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"Pseudoinverse" redirects here. For the Moore–Penrose pseudoinverse, sometimes referred to as "the pseudoinverse", see Moore–Penrose pseudoinverse.

In mathematics, a generalized inverse of a matrix A is a matrix that has some properties of the inverse matrix of A but not necessarily all of them. Formally, given a matrix A \in \mathbb{R}^{n\times m} and a matrix A^{\mathrm g} \in \mathbb{R}^{m\times n}, A^{\mathrm g} is a generalized inverse of A if it satisfies the condition  AA^{\mathrm g}A = A.

The purpose of constructing a generalized inverse is to obtain a matrix that can serve as the inverse in some sense for a wider class of matrices than invertible ones. Typically, the generalized inverse exists for an arbitrary matrix, and when a matrix has an inverse, then its inverse and the generalized inverse are the same. Some generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup.

Types of generalized inverses[edit]

The Penrose conditions are used to define different generalized inverses: for A \in \mathbb{R}^{n\times m} and A^{\mathrm g} \in \mathbb{R}^{m\times n},

1.) AA^{\mathrm g}A = A
2.) A^{\mathrm g}AA^{\mathrm g}= A^{\mathrm g}
3.) (AA^{\mathrm g})^{\mathrm T} = AA^{\mathrm g}
4.) (A^{\mathrm g}A)^{\mathrm T} = A^{\mathrm g}A .

If A^{\mathrm g} satisfies condition (1.), it is a generalized inverse of A, if it satisfies conditions (1.) and (2.) then it is a generalized reflexive inverse of A, and if it satisfies all 4 conditions, then it is a Moore–Penrose pseudoinverse of A.

Other various kinds of generalized inverses include


Any generalized inverse can be used to determine if a system of linear equations has any solutions, and if so to give all of them.[1] If any solutions exist for the n × m linear system


with vector x of unknowns and vector b of constants, all solutions are given by

x=A^{\mathrm g}b + [I-A^{\mathrm g}A]w

parametric on the arbitrary vector w, where A^{\mathrm g} is any generalized inverse of A. Solutions exist if and only if A^{\mathrm g}b is a solution – that is, if and only if AA^{\mathrm g}b=b.

See also[edit]


  1. ^ James, M., "The generalised inverse", Mathematical Gazette 62, June 1978, 109–114.
  • Yoshihiko Nakamura (1991). * Advanced Robotics: Redundancy and Optimization. Addison-Wesley. ISBN 0201151987. 
  • Zheng, B; Bapat, R. B. (2004). "Generalized inverse A(2)T,S and a rank equation". Applied Mathematics and Computation 155: 407–415. doi:10.1016/S0096-3003(03)00786-0. 
  • S. L. Campbell and C. D. Meyer (1991). Generalized Inverses of Linear Transformations. Dover. ISBN 978-0-486-66693-8. 
  • Adi Ben-Israel and Thomas N.E. Greville (2003). Generalized inverses. Theory and applications (2nd ed.). New York, NY: Springer. ISBN 0-387-00293-6. 
  • C. R. Rao and C. Radhakrishna Rao and Sujit Kumar Mitra (1971). Generalized Inverse of Matrices and its Applications. New York: John Wiley & Sons. p. 240. ISBN 0-471-70821-6. 

External links[edit]