Generalized inverse

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.

Motivation for the generalized inverse

Consider the linear system

${\displaystyle Ax=y}$

where $A$ is an $n\times m$ matrix and ${\displaystyle y\in {\mathcal {R}}(A)}$, the range space of $A$. If the matrix $A$ is nonsingular then ${\displaystyle x=A^{-1}y}$ will be the solution of the system. Note that, if a matrix $A$ is nonsingular

${\displaystyle AA^{-1}A=A.}$

Suppose the matrix $A$ is singular or ${\displaystyle n\neq m}$ then we need a right candidate $G$ of order $m\times n$ such that for all $y$,

${\displaystyle AGy=y.}$

That is ${\displaystyle Gy}$ is a solution of the linear system $Ax=y$. Equivalently, $G$ of order $m\times n$ such that

${\displaystyle AGA=A.}$

Hence we can define the generalized inverse as follows: Given a $n\times m$ matrix $A$, a $m\times n$ matrix $G$ is said to be generalized inverse of $A$ if ${\displaystyle AGA=A.}$

Construction of generalized inverse

^[1]

The following characterizations are easy to verify.

1. If ${\displaystyle A=BC}$ is a rank factorization, then ${\displaystyle G=C_{r}^{-}B_{l}^{-}}$ is a g-inverse of $A$ where ${\displaystyle C_{r}^{-}}$ is a right inverse of $C$ and ${\displaystyle B_{l}^{-}}$ is left inverse of $B$.
2. If ${\displaystyle A=P{\begin{bmatrix}I_{r}&0\\0&0\end{bmatrix}}Q}$ for any non-singular matrices $P$ and $Q$, then ${\displaystyle G=Q^{-1}{\begin{bmatrix}I_{r}&U\\W&V\end{bmatrix}}P^{-1}}$ is a generalized inverse of $A$ for arbitrary $U,V$ and $W$.
3. Let $A$ be of rank $r$. Without loss of generality, let
   

${\displaystyle A={\begin{bmatrix}B&C\\D&E\end{bmatrix}}.}$

where ${\displaystyle B_{r\times r}}$ is the non-singular submatrix of $A$. Then,

${\displaystyle G={\begin{bmatrix}B^{-1}&0\\0&0\end{bmatrix}}}$ is a g-inverse of $A$.

Types of generalized inverses

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

If ${\displaystyle A^{\mathrm {g} }}$ satisfies the first condition, then it is a generalized inverse of $A$. If it satisfies the first two conditions, then it is a generalized reflexive inverse of $A$. If it satisfies all four conditions, then it is a Moore–Penrose pseudoinverse of $A$.

Other various kinds of generalized inverses include

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

Uses

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.^[2] If any solutions exist for the n × m linear system

$Ax=b$

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

• Inverse element

References

1. Bapat, Ravindra B. Linear algebra and linear models. Springer Science & Business Media, 2012.springer.com/book
2. 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

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