Generalized inverse

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

In mathematics, a generalized inverse or pseudoinverse of a matrix A is a matrix that has some properties of the inverse matrix of A but not necessarily all of them. The term "the pseudoinverse" commonly means the Moore–Penrose pseudoinverse.

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

The various kinds of generalized inverses include

• One-sided inverse (left inverse or right inverse) If the matrix A has dimensions $m \times n$ then use the left inverse if m > n and the right inverse if m < n
  • Left inverse is given by $A_{\mathrm{left}}^{-1} = \left(A^T A\right)^{-1} A^T$, i.e. $A_{\mathrm{left}}^{-1} A = I_n$ where I_n is the $n \times n$ identity matrix.
  • Right inverse is given by $A_{\mathrm{right}}^{-1} = A^T \left(A A^T\right)^{-1}$, i.e. $A A_{\mathrm{right}}^{-1} = I_m$ where I_m is the $m \times m$ identity matrix.
• Drazin inverse
• Bott–Duffin inverse
• Moore–Penrose pseudoinverse

See also

• Inverse element

References

• 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 978-0486666938. 
• 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. http://www.zentralblatt-math.org/zmath/en/search/?q=an:1026.15004&format=complete. 
• C. R. Rao and C. Radhakrishna Rao and Sujit Kumar Mitra (1971). Generalized Inverse of Matrices and its Applications. New York: John Wiley & Sons. pp. 240. ISBN 0-471-70821-6. 

External links

• 15A09 Matrix inversion, generalized inverses in Mathematics Subject Classification, MathSciNet search