Graphics Reference
In-Depth Information
Figure 1.20.
Computing the generalized
inverse for Example 1.11.2.
y
b
x
u
a
ker(T)
L = ker(T)
T
-1
(b)
1.11.2. Example.
Consider the map T :
R
2
Æ
R
defined by T(x,y) = x - y. Let b Œ
R
.
We want to show that the generalized inverse T
+
:
R
Æ
R
2
is defined by
Tb
b
+
()
=-
(
)
11
,
.
2
Solution.
See Figure 1.20. The kernel of T, ker(T), is the line x = y in
R
2
. The ortho-
gonal complement of ker(T) is the line
L
defined by x + y = 0. If
a
= T
+
(b), then
a
is
the point where the line T
-1
(b) meets
L
. Clearly, such a point
a
is just the orthogonal
projection of the vector (b,0) on
L
, that is,
b
=
(
(
)
)
(
)
au
b
,
0
u
=-
11
,
,
2
for any unit direction vector
u
for
L
(Theorems 4.5.12 and 1.4.6). For example, we
could choose
1
2
(
)
u
=
11
,
-
.
Of interest to us is the matrix version of the generalized inverse. Let A be an
arbitrary real m ¥ n matrix. Let T :
R
m
Æ
R
n
be the natural linear transformation
associated to this matrix by the formula T(
x
) =
x
A.
Definition.
The n ¥ m matrix A
+
for the generalized inverse T
+
is called the
general-
ized inverse
or
pseudo-inverse
or
Moore-Penrose inverse matrix
for A.
1.11.3. Theorem.
(1) The generalized inverse matrix A
+
for a matrix A satisfies
T
T
(
)
(
)
+
+
+
+
+
+
+
+
AA A
=
A
,
A AA
=
A
,
A A
=
A A
,
and
AA
=
AA
.
