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

$Ax=y$

where $A$ is an $n\times m$ matrix and $y\in \mathcal R(A)$, the range 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

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

Suppose the matrix $A$ is singular or $n\neq m$ then we need a right candidate $G$ of order $m\times n$ such that

$AGy=y.$

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

$AGA=A.$

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 $AGA=A.$

Construction of generalized inverse

^[1]

The following characterizations are easy to verify.

1. 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 $B$.
2. 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 $W$.
3. 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 $A^{\mathrm g} \in \mathbb{R}^{m\times n},$

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

• 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 $A_{\mathrm{right}}^{-1} = A^{\mathrm T} \left(A A^{\mathrm T}\right)^{-1}$, i.e. $A A_{\mathrm{right}}^{-1} = I_n$ 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