17 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Generalized inverse | 3/3 | https://en.wikipedia.org/wiki/Generalized_inverse | reference | science, encyclopedia | 2026-05-05T07:23:55.090630+00:00 | kb-cron |
A right inverse of a non-square matrix
A
{\displaystyle A}
is given by
A
R
−
1
=
A
⊺
(
A
A
⊺
)
−
1
{\displaystyle A_{\mathrm {R} }^{-1}=A^{\intercal }\left(AA^{\intercal }\right)^{-1}}
, provided
A
{\displaystyle A}
has full row rank. A left inverse of a non-square matrix
A
{\displaystyle A}
is given by
A
L
−
1
=
(
A
⊺
A
)
−
1
A
⊺
{\displaystyle A_{\mathrm {L} }^{-1}=\left(A^{\intercal }A\right)^{-1}A^{\intercal }}
, provided
A
{\displaystyle A}
has full column rank. If
A
=
B
C
{\displaystyle A=BC}
is a rank factorization, then
G
=
C
R
−
1
B
L
−
1
{\displaystyle G=C_{\mathrm {R} }^{-1}B_{\mathrm {L} }^{-1}}
is a g-inverse of
A
{\displaystyle A}
, where
C
R
−
1
{\displaystyle C_{\mathrm {R} }^{-1}}
is a right inverse of
C
{\displaystyle C}
and
B
L
−
1
{\displaystyle B_{\mathrm {L} }^{-1}}
is left inverse of
B
{\displaystyle B}
. If
A
=
P
[
I
r
0
0
0
]
Q
{\displaystyle A=P{\begin{bmatrix}I_{r}&0\\0&0\end{bmatrix}}Q}
for any non-singular matrices
P
{\displaystyle P}
and
Q
{\displaystyle Q}
, then
G
=
Q
−
1
[
I
r
U
W
V
]
P
−
1
{\displaystyle G=Q^{-1}{\begin{bmatrix}I_{r}&U\\W&V\end{bmatrix}}P^{-1}}
is a generalized inverse of
A
{\displaystyle A}
for arbitrary
U
,
V
{\displaystyle U,V}
and
W
{\displaystyle W}
. Let
A
{\displaystyle A}
be of rank
r
{\displaystyle r}
. Without loss of generality, let
A
=
[
B
C
D
E
]
,
{\displaystyle A={\begin{bmatrix}B&C\\D&E\end{bmatrix}},}
where
B
r
×
r
{\displaystyle B_{r\times r}}
is the non-singular submatrix of
A
{\displaystyle A}
. Then,
G
=
[
B
−
1
0
0
0
]
{\displaystyle G={\begin{bmatrix}B^{-1}&0\\0&0\end{bmatrix}}}
is a generalized inverse of
A
{\displaystyle A}
if and only if
E
=
D
B
−
1
C
{\displaystyle E=DB^{-1}C}
.
== 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
A
x
=
b
,
{\displaystyle Ax=b,}
with vector
x
{\displaystyle x}
of unknowns and vector
b
{\displaystyle b}
of constants, all solutions are given by
x
=
A
g
b
+
[
I
−
A
g
A
]
w
,
{\displaystyle x=A^{\mathrm {g} }b+\left[I-A^{\mathrm {g} }A\right]w,}
parametric on the arbitrary vector
w
{\displaystyle w}
, where
A
g
{\displaystyle A^{\mathrm {g} }}
is any generalized inverse of
A
{\displaystyle A}
. Solutions exist if and only if
A
g
b
{\displaystyle A^{\mathrm {g} }b}
is a solution, that is, if and only if
A
A
g
b
=
b
{\displaystyle AA^{\mathrm {g} }b=b}
. If A has full column rank, the bracketed expression in this equation is the zero matrix and so the solution is unique.
== Generalized inverses of matrices == The generalized inverses of matrices can be characterized as follows. Let
A
∈
R
m
×
n
{\displaystyle A\in \mathbb {R} ^{m\times n}}
, and
A
=
U
[
Σ
1
0
0
0
]
V
T
{\displaystyle A=U{\begin{bmatrix}\Sigma _{1}&0\\0&0\end{bmatrix}}V^{\operatorname {T} }}
be its singular-value decomposition. Then for any generalized inverse
A
g
{\displaystyle A^{g}}
, there exist matrices
X
{\displaystyle X}
,
Y
{\displaystyle Y}
, and
Z
{\displaystyle Z}
such that
A
g
=
V
[
Σ
1
−
1
X
Y
Z
]
U
T
.
{\displaystyle A^{g}=V{\begin{bmatrix}\Sigma _{1}^{-1}&X\\Y&Z\end{bmatrix}}U^{\operatorname {T} }.}
Conversely, any choice of
X
{\displaystyle X}
,
Y
{\displaystyle Y}
, and
Z
{\displaystyle Z}
for matrix of this form is a generalized inverse of
A
{\displaystyle A}
. The
{
1
,
2
}
{\displaystyle \{1,2\}}
-inverses are exactly those for which
Z
=
Y
Σ
1
X
{\displaystyle Z=Y\Sigma _{1}X}
, the
{
1
,
3
}
{\displaystyle \{1,3\}}
-inverses are exactly those for which
X
=
0
{\displaystyle X=0}
, and the
{
1
,
4
}
{\displaystyle \{1,4\}}
-inverses are exactly those for which
Y
=
0
{\displaystyle Y=0}
. In particular, the pseudoinverse is given by
X
=
Y
=
Z
=
0
{\displaystyle X=Y=Z=0}
:
A
+
=
V
[
Σ
1
−
1
0
0
0
]
U
T
.
{\displaystyle A^{+}=V{\begin{bmatrix}\Sigma _{1}^{-1}&0\\0&0\end{bmatrix}}U^{\operatorname {T} }.}
== Transformation consistency properties == In practical applications it is necessary to identify the class of matrix transformations that must be preserved by a generalized inverse. For example, the Moore–Penrose inverse,
A
+
,
{\displaystyle A^{+},}
satisfies the following definition of consistency with respect to transformations involving unitary matrices U and V:
(
U
A
V
)
+
=
V
∗
A
+
U
∗
{\displaystyle (UAV)^{+}=V^{*}A^{+}U^{*}}
. The Drazin inverse,
A
D
{\displaystyle A^{\mathrm {D} }}
satisfies the following definition of consistency with respect to similarity transformations involving a nonsingular matrix S:
(
S
A
S
−
1
)
D
=
S
A
D
S
−
1
{\displaystyle \left(SAS^{-1}\right)^{\mathrm {D} }=SA^{\mathrm {D} }S^{-1}}
. The unit-consistent (UC) inverse,
A
U
,
{\displaystyle A^{\mathrm {U} },}
satisfies the following definition of consistency with respect to transformations involving nonsingular diagonal matrices D and E:
(
D
A
E
)
U
=
E
−
1
A
U
D
−
1
{\displaystyle (DAE)^{\mathrm {U} }=E^{-1}A^{\mathrm {U} }D^{-1}}
. The fact that the Moore–Penrose inverse provides consistency with respect to rotations (which are orthonormal transformations) explains its widespread use in physics and other applications in which Euclidean distances must be preserved. The UC inverse, by contrast, is applicable when system behavior is expected to be invariant with respect to the choice of units on different state variables, e.g., miles versus kilometers.
== See also == Block matrix pseudoinverse Regular semigroup
== Citations ==
== Sources ==
=== Textbook === Ben-Israel, Adi; Greville, Thomas Nall Eden (2003). Generalized Inverses: Theory and Applications (2nd ed.). New York, NY: Springer. doi:10.1007/b97366. ISBN 978-0-387-00293-4. Campbell, Stephen L.; Meyer, Carl D. (1991). Generalized Inverses of Linear Transformations. Dover. ISBN 978-0-486-66693-8. Horn, Roger Alan; Johnson, Charles Royal (1985). Matrix Analysis. Cambridge University Press. ISBN 978-0-521-38632-6. Nakamura, Yoshihiko (1991). Advanced Robotics: Redundancy and Optimization. Addison-Wesley. ISBN 978-0201151985. Rao, C. Radhakrishna; Mitra, Sujit Kumar (1971). Generalized Inverse of Matrices and its Applications. New York: John Wiley & Sons. pp. 240. ISBN 978-0-471-70821-6.
=== Publication === James, M. (June 1978). "The generalised inverse". The Mathematical Gazette. 62 (420): 109–114. doi:10.2307/3617665. JSTOR 3617665. Uhlmann, Jeffrey K. (2018). "A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations". SIAM Journal on Matrix Analysis and Applications. 239 (2): 781–800. doi:10.1137/17M113890X. Zheng, Bing; Bapat, Ravindra (2004). "Generalized inverse A(2)T,S and a rank equation". Applied Mathematics and Computation. 155 (2): 407–415. doi:10.1016/S0096-3003(03)00786-0.