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---
title: "Generalized inverse"
chunk: 2/3
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"
---
(
A
A
g
)
∗
=
A
A
g
{\displaystyle (AA^{\mathrm {g} })^{*}=AA^{\mathrm {g} }}
(
A
g
A
)
∗
=
A
g
A
,
{\displaystyle (A^{\mathrm {g} }A)^{*}=A^{\mathrm {g} }A,}
where
∗
{\displaystyle {}^{*}}
denotes conjugate transpose. If
A
g
{\displaystyle A^{\mathrm {g} }}
satisfies the first condition, then it is a generalized inverse of
A
{\displaystyle A}
. If it satisfies the first two conditions, then it is a reflexive generalized inverse of
A
{\displaystyle A}
. If it satisfies all four conditions, then it is the pseudoinverse of
A
{\displaystyle A}
, which is denoted by
A
+
{\displaystyle A^{+}}
and also known as the Moore–Penrose inverse, after the pioneering works by E. H. Moore and Roger Penrose. It is convenient to define an
I
{\displaystyle I}
-inverse of
A
{\displaystyle A}
as an inverse that satisfies the subset
I
⊂
{
1
,
2
,
3
,
4
}
{\displaystyle I\subset \{1,2,3,4\}}
of the Penrose conditions listed above. Relations, such as
A
(
1
,
4
)
A
A
(
1
,
3
)
=
A
+
{\displaystyle A^{(1,4)}AA^{(1,3)}=A^{+}}
, can be established between these different classes of
I
{\displaystyle I}
-inverses.
When
A
{\displaystyle A}
is non-singular, any generalized inverse
A
g
{\displaystyle A^{\mathrm {g} }}
is equal to
A
−
1
{\displaystyle A^{-1}}
and is therefore unique. For a singular
A
{\displaystyle A}
, some generalized inverses, such as the Drazin inverse and the Moore–Penrose inverse, are unique, while others are not necessarily uniquely defined.
== Examples ==
=== Non-reflexive generalized inverse ===
Let
A
=
[
1
0
3
2
0
6
0
0
0
]
,
G
=
[
1
0
0
0
0
1
0
0
0
]
.
{\displaystyle A={\begin{bmatrix}1&0&3\\2&0&6\\0&0&0\end{bmatrix}},\quad G={\begin{bmatrix}1&0&0\\0&0&1\\0&0&0\end{bmatrix}}.}
Obviously,
A
{\displaystyle A}
is singular.
A
{\displaystyle A}
and
G
{\displaystyle G}
satisfy Penrose conditions (1), but not the other there. Hence,
G
{\displaystyle G}
is a non-reflexive generalized inverse of
A
{\displaystyle A}
.
The first column of
A
{\displaystyle A}
spans
im
⁡
A
{\displaystyle \operatorname {im} A}
, and
G
{\displaystyle G}
maps it to
(
1
,
0
,
0
)
{\displaystyle (1,0,0)}
, which does not lie in
ker
⁡
A
{\displaystyle \ker A}
. Additionally,
G
{\displaystyle G}
maps
(
0
,
0
,
1
)
{\displaystyle (0,0,1)}
to
(
0
,
1
,
0
)
{\displaystyle (0,1,0)}
, which lies in
ker
⁡
A
{\displaystyle \ker A}
. The relationship is summarized in the picture on the right.
=== Reflexive generalized inverse ===
Let
A
=
[
1
2
3
4
5
6
7
8
9
]
,
G
=
[
−
5
3
2
3
0
4
3
−
1
3
0
0
0
0
]
.
{\displaystyle A={\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}},\quad G={\begin{bmatrix}-{\frac {5}{3}}&{\frac {2}{3}}&0\\[4pt]{\frac {4}{3}}&-{\frac {1}{3}}&0\\[4pt]0&0&0\end{bmatrix}}.}
Since
det
(
A
)
=
0
{\displaystyle \det(A)=0}
,
A
{\displaystyle A}
is singular and has no regular inverse. However,
A
{\displaystyle A}
and
G
{\displaystyle G}
satisfy Penrose conditions (1) and (2), but not (3) or (4). Hence,
G
{\displaystyle G}
is a reflexive generalized inverse of
A
{\displaystyle A}
.
=== One-sided inverse ===
Let
A
=
[
1
2
3
4
5
6
]
,
A
R
−
1
=
[
−
17
18
8
18
−
2
18
2
18
13
18
−
4
18
]
.
{\displaystyle A={\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}},\quad A_{\mathrm {R} }^{-1}={\begin{bmatrix}-{\frac {17}{18}}&{\frac {8}{18}}\\[4pt]-{\frac {2}{18}}&{\frac {2}{18}}\\[4pt]{\frac {13}{18}}&-{\frac {4}{18}}\end{bmatrix}}.}
Since
A
{\displaystyle A}
is not square,
A
{\displaystyle A}
has no regular inverse. However,
A
R
−
1
{\displaystyle A_{\mathrm {R} }^{-1}}
is a right inverse of
A
{\displaystyle A}
. The matrix
A
{\displaystyle A}
has no left inverse.
=== Inverse of other semigroups (or rings) ===
The element b is a generalized inverse of an element a if and only if
a
⋅
b
⋅
a
=
a
{\displaystyle a\cdot b\cdot a=a}
, in any semigroup (or ring, since the multiplication function in any ring is a semigroup).
The generalized inverses of the element 3 in the ring
Z
/
12
Z
{\displaystyle \mathbb {Z} /12\mathbb {Z} }
are 3, 7, and 11, since in the ring
Z
/
12
Z
{\displaystyle \mathbb {Z} /12\mathbb {Z} }
:
3
⋅
3
⋅
3
=
3
{\displaystyle 3\cdot 3\cdot 3=3}
3
⋅
7
⋅
3
=
3
{\displaystyle 3\cdot 7\cdot 3=3}
3
⋅
11
⋅
3
=
3
{\displaystyle 3\cdot 11\cdot 3=3}
The generalized inverses of the element 4 in the ring
Z
/
12
Z
{\displaystyle \mathbb {Z} /12\mathbb {Z} }
are 1, 4, 7, and 10, since in the ring
Z
/
12
Z
{\displaystyle \mathbb {Z} /12\mathbb {Z} }
:
4
⋅
1
⋅
4
=
4
{\displaystyle 4\cdot 1\cdot 4=4}
4
⋅
4
⋅
4
=
4
{\displaystyle 4\cdot 4\cdot 4=4}
4
⋅
7
⋅
4
=
4
{\displaystyle 4\cdot 7\cdot 4=4}
4
⋅
10
⋅
4
=
4
{\displaystyle 4\cdot 10\cdot 4=4}
If an element a in a semigroup (or ring) has an inverse, the inverse must be the only generalized inverse of this element, like the elements 1, 5, 7, and 11 in the ring
Z
/
12
Z
{\displaystyle \mathbb {Z} /12\mathbb {Z} }
.
In the ring
Z
/
12
Z
{\displaystyle \mathbb {Z} /12\mathbb {Z} }
any element is a generalized inverse of 0; however 2 has no generalized inverse, since there is no b in
Z
/
12
Z
{\displaystyle \mathbb {Z} /12\mathbb {Z} }
such that
2
⋅
b
⋅
2
=
2
{\displaystyle 2\cdot b\cdot 2=2}
.
== Construction ==
The following characterizations are easy to verify:
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