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(2) The generalized inverse matrix is
uniquely
defined by the identities in (1), that
is, if G is a matrix satisfying
T
T
(
)
(
)
AGA
=
A
,
GAG
=
G
,
GA
=
GA
,
and
AG
=
AG
,
then G = A
+
.
Proof.
See [Penr55] or [RaoM71].
1.11.4. Corollary.
(1) If A is a real m ¥ n matrix of rank n, then A
+
= (A
T
A)
-1
A
T
.
(2) If A is a real m ¥ n matrix of rank m, then A
+
= A
T
(AA
T
)
-1
.
Proof.
The Corollary follows from Theorem 1.11.3(2). For part (1), it is easy to check
that A
T
A is a nonsingular n ¥ n matrix and (A
T
A)
-1
A
T
satisfies the stated identities.
Part (2) follows from a similar argument.
To compute the matrix A
+
for the map T
+
in Example 1.11.2 above.
1.11.5. Example.
In this case, we have that A
T
= (1 -1), so that A
T
A = 2 and
Solution.
1
2
-
1
=
(
)
+
T
T
(
)
AAAA
=
11,
-
which agrees with our formula for T
+
.
A nice application of the generalized inverse matrix and Corollary 1.11.4 is to a
linear least squares approximation problem. Suppose that we are given a real m ¥ n
matrix A with n > m and
b
Œ
R
n
. We want to solve the equation
xb
A =
(1.27)
for
x
Œ
R
m
. Unfortunately, the system of equations defined by (1.27) is overdetermined
and may not have a solution. The best that we can do in general is to solve the
following problem:
A linear least squares approximation problem:
Given an m ¥ n matrix A with n > m
and
b
Œ
R
n
, find a point
a
0
Œ
R
m
that minimizes the distances |
a
A -
b
|, that is, find
a
0
so
that
{
}
(1.28)
a
0
Ab
-=
min
a
A
-
b
.
m
aR
Œ
It is easy to explain the name of the problem. Let
=
()
=
(
)
=
(
)
A
a
,
a
a
,
a
,...,
a
,
and
b
b
,
b
,...,
b
.
ij
12
m
12
n
