Graphics Reference
In-Depth Information
1.11
The Generalized Inverse Matrix
Let
mn
T
:
RR
Æ
be a linear transformation. Now normally one would not expect this arbitrary map T
to have an inverse, especially if m > n, but it turns out that it is possible to define
something close to that that is useful. Define a map
+
n
m
T
:
RR
Æ
as follows: See Figure 1.19. Let
b
Œ
R
n
. The point
b
may not be in the image of T,
im(T), since we are not assuming that T is onto, but im(T) is a plane in
R
n
. There-
fore, there is a unique point
c
Œ im(T) that is closest to
b
(Theorem 4.5.12). If the
transformation T is onto, then obviously
c
=
b
. It is easy to show that T
-1
(
c
) is a plane
in
R
m
that is parallel to the kernel of T, ker(T). This plane will meet the orthogonal
complement of the kernel of T, ker(T)
^
, in a unique point
a
. For an alternative
definition of the point
a
write
R
m
in the form
()
^
R
m
()
≈
=
ker T
ker T
and let
^
^
()
()
Æ
()
j=
T
ker T
: ker T
im T
.
It is easy to show that j is an isomorphism and
a
=j
-1
(
c
). In either case, we define
T
+
(
b
) =
a
.
The map T
+
is called the
generalized
or
Moore-Penrose inverse
of T.
Definition.
T
+
is a well-defined linear transformation.
1.11.1. Lemma.
Proof.
Easy.
ker(T)
R
m
b
R
n
T
-1
(c)
c
T
a
im(T)
j
j
-1
ker(T)
Figure 1.19.
The geometry behind the generalized inverse.