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---
title: "Generalized inverse"
chunk: 1/3
source: "https://en.wikipedia.org/wiki/Generalized_inverse"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T07:23:55.090630+00:00"
instance: "kb-cron"
---
In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element x is an element y that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized inverse of a matrix is to obtain a matrix that can serve as an inverse in some sense for a wider class of matrices than invertible matrices. Generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup. This article describes generalized inverses of a matrix
A
{\displaystyle A}
.
A matrix
A
g
∈
R
n
×
m
{\displaystyle A^{\mathrm {g} }\in \mathbb {R} ^{n\times m}}
is a generalized inverse of a matrix
A
∈
R
m
×
n
{\displaystyle A\in \mathbb {R} ^{m\times n}}
if
A
A
g
A
=
A
.
{\displaystyle AA^{\mathrm {g} }A=A.}
A generalized inverse exists for an arbitrary matrix, and when a matrix has a regular inverse, this inverse is its unique generalized inverse.
== Motivation ==
Consider the linear system
A
x
=
y
{\displaystyle Ax=y}
where
A
{\displaystyle A}
is an
m
×
n
{\displaystyle m\times n}
matrix and
y
∈
C
(
A
)
,
{\displaystyle y\in {\mathcal {C}}(A),}
the column space of
A
{\displaystyle A}
. If
m
=
n
{\displaystyle m=n}
and
A
{\displaystyle A}
is nonsingular then
x
=
A
−
1
y
{\displaystyle x=A^{-1}y}
will be the solution of the system. Note that, if
A
{\displaystyle A}
is nonsingular, then
A
A
−
1
A
=
A
.
{\displaystyle AA^{-1}A=A.}
Now suppose
A
{\displaystyle A}
is rectangular (
m
≠
n
{\displaystyle m\neq n}
), or square and singular. Then we need a right candidate
G
{\displaystyle G}
of order
n
×
m
{\displaystyle n\times m}
such that for all
y
∈
C
(
A
)
,
{\displaystyle y\in {\mathcal {C}}(A),}
A
G
y
=
y
.
{\displaystyle AGy=y.}
That is,
x
=
G
y
{\displaystyle x=Gy}
is a solution of the linear system
A
x
=
y
{\displaystyle Ax=y}
.
Equivalently, we need a matrix
G
{\displaystyle G}
of order
n
×
m
{\displaystyle n\times m}
such that
A
G
A
=
A
.
{\displaystyle AGA=A.}
Hence we can define the generalized inverse as follows: Given an
m
×
n
{\displaystyle m\times n}
matrix
A
{\displaystyle A}
, an
n
×
m
{\displaystyle n\times m}
matrix
G
{\displaystyle G}
is said to be a generalized inverse of
A
{\displaystyle A}
if
A
G
A
=
A
.
{\displaystyle AGA=A.}
‍ The matrix
A
−
1
{\displaystyle A^{-1}}
has been termed a regular inverse of
A
{\displaystyle A}
by some authors.
The problem is how to choose an
x
{\displaystyle x}
as the output of
G
{\displaystyle G}
for every
y
{\displaystyle y}
when the map
x
↦
y
=
A
x
{\displaystyle x\mapsto y=Ax}
is not bijective.
If
A
{\displaystyle A}
is not surjective, then not all
y
{\displaystyle y}
's in its codomain have corresponding
x
{\displaystyle x}
's via
A
{\displaystyle A}
. To circumvent it, we just let
G
{\displaystyle G}
map those
y
{\displaystyle y}
's to arbitrary values.
For example, decompose the codomain of
A
{\displaystyle A}
as the direct sum of the column space
C
(
A
)
{\displaystyle {\mathcal {C}}(A)}
and a complement subspace, and construct
G
{\displaystyle G}
as follows.
For
y
{\displaystyle y}
's in the former subspace, let
G
{\displaystyle G}
map back to the corresponding
x
{\displaystyle x}
's.
For
y
{\displaystyle y}
's in the latter subspace, let
G
{\displaystyle G}
map them all to zero (as there're no corresponding
x
{\displaystyle x}
's).
For other
y
{\displaystyle y}
's, decompose them as the sum of the above two components, apply
G
{\displaystyle G}
respectively, then take the sum.
If
A
{\displaystyle A}
is not injective, then some
y
{\displaystyle y}
's correspond to multiple
x
{\displaystyle x}
's via
A
{\displaystyle A}
. To circumvent it, we let
G
{\displaystyle G}
map every
y
{\displaystyle y}
to one of the
x
{\displaystyle x}
's according an algorithm.
For example, decompose the domain of
A
{\displaystyle A}
as the direct sum of
ker
⁡
A
{\displaystyle \ker A}
and a complement subspace. For every possible
y
{\displaystyle y}
, its preimage must be parallel to
ker
⁡
A
{\displaystyle \ker A}
and intersect the chosen complement subspace at a single point. Let
G
{\displaystyle G}
map the
y
{\displaystyle y}
to this point.
If
A
{\displaystyle A}
is neither surjective nor injective, we combine the above two tricks.
The picture on the right is an example.
== Types ==
Important types of generalized inverse include:
One-sided inverse (right inverse or left inverse)
Right inverse: If the matrix
A
{\displaystyle A}
has dimensions
m
×
n
{\displaystyle m\times n}
and
rank
(
A
)
=
m
{\displaystyle {\textrm {rank}}(A)=m}
, then there exists an
n
×
m
{\displaystyle n\times m}
matrix
A
R
−
1
{\displaystyle A_{\mathrm {R} }^{-1}}
called the right inverse of
A
{\displaystyle A}
such that
A
A
R
−
1
=
I
m
{\displaystyle AA_{\mathrm {R} }^{-1}=I_{m}}
, where
I
m
{\displaystyle I_{m}}
is the
m
×
m
{\displaystyle m\times m}
identity matrix.
Left inverse: If the matrix
A
{\displaystyle A}
has dimensions
m
×
n
{\displaystyle m\times n}
and
rank
(
A
)
=
n
{\displaystyle {\textrm {rank}}(A)=n}
, then there exists an
n
×
m
{\displaystyle n\times m}
matrix
A
L
−
1
{\displaystyle A_{\mathrm {L} }^{-1}}
called the left inverse of
A
{\displaystyle A}
such that
A
L
−
1
A
=
I
n
{\displaystyle A_{\mathrm {L} }^{-1}A=I_{n}}
, where
I
n
{\displaystyle I_{n}}
is the
n
×
n
{\displaystyle n\times n}
identity matrix.
Bott–Duffin inverse
Drazin inverse
Moore–Penrose inverse
Some generalized inverses are defined and classified based on the Penrose conditions:
A
A
g
A
=
A
{\displaystyle AA^{\mathrm {g} }A=A}
A
g
A
A
g
=
A
g
{\displaystyle A^{\mathrm {g} }AA^{\mathrm {g} }=A^{\mathrm {g} }}
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