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 ΠR 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 Πim(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.
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