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 matrices. 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

Ax=y

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

AA^{-1}A=A.

Suppose the matrix $A$ is singular, or $n\neq m$ . Then we need a right candidate $G$ of order $m\times n$ such that for all $y\in {\mathcal {R}}(A)$ ,

AGy=y.

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

AGA=A.

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

Construction of generalized inverse

The following characterizations are easy to verify:

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

A={\begin{bmatrix}B&C\\D&E\end{bmatrix}},

where

B_{r\times r}

is the non-singular submatrix of

A

. Then,

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 $A^{{{\mathrm g}}}\in {\mathbb {R}}^{{m\times n}}$ :

$AA^{\mathrm {g} }A=A$
$A^{\mathrm {g} }AA^{\mathrm {g} }=A^{\mathrm {g} }$
$(AA^{\mathrm {g} })^{\mathrm {T} }=AA^{\mathrm {g} }$
$(A^{\mathrm {g} }A)^{\mathrm {T} }=A^{\mathrm {g} }A$

If $A^{\mathrm {g} }$ 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 kinds of generalized inverse 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 $A_{\mathrm {left} }^{-1}=\left(A^{\mathrm {T} }A\right)^{-1}A^{\mathrm {T} }$ , i.e., $A_{{{\mathrm {left}}}}^{{-1}}A=I_{m}$ , where $I_m$ is the $m\times m$ identity matrix.
- Right inverse is given by $A_{\mathrm {right} }^{-1}=A^{\mathrm {T} }\left(AA^{\mathrm {T} }\right)^{-1}$ , 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 whether 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

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 $AA^{\mathrm {g} }b=b$ .

Notes

↑ James, M. (June 1978). "The generalised inverse". Mathematical Gazette. 62: 109–114. doi:10.2307/3617665.

References

Ben-Israel, Adi; Greville, Thomas N.E. (2003). Generalized inverses. Theory and applications (2nd ed.). New York, NY: Springer. ISBN 0-387-00293-6.
Campbell, S. L.; Meyer, C. D. (1991). Generalized Inverses of Linear Transformations. Dover. ISBN 978-0-486-66693-8.
Nakamura, Yoshihiko (1991). * Advanced Robotics: Redundancy and Optimization. Addison-Wesley. ISBN 0201151987.
Rao, C. R.; Rao, C. Radhakrishna; Mitra, Sujit Kumar (1971). Generalized Inverse of Matrices and its Applications. New York: John Wiley & Sons. p. 240. ISBN 0-471-70821-6.
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.

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