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 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:

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 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. 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

• 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.