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. A generalized inverse exists for an arbitrary matrix, and when a matrix has an inverse, then this inverse is its unique generalized inverse. 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. (June 1978). "The generalised inverse". Mathematical Gazette 62: 109–114. doi:10.2307/3617665. 
  • 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]