If one understands the concept of a null space, the left null space is
extremely easy to understand.
- Definition 1:
Left Null Space
The Left Null Space of a matrix is the
null space of its
transpose,
i.e.,
𝒩AT={y∈
ℝ
m
|ATy=0}
𝒩
A
A
y
0
y
ℝ
m
The word "left" in this context stems from the fact that
ATy=0
A
y
0
is equivalent to
yTA=0
y
A
0
where
yy "acts" on
AA from the left.
As
A
red
A
red
was the key to identifying the
null space of
AA, we shall see that
A
red
T
A
red
T
is the key to the null space of
AT
A
.
If
A=111213
A
11
12
13
(1)
then
AT=111123
A
111
123
(2)
and so
A
red
T
=111012
A
red
T
111
012
(3)
We solve
A
red
T
=0
A
red
T
0
by recognizing that
y
1
y
1
and
y
2
y
2
are pivot variables while
y
3
y
3
is free. Solving
A
red
T
y=0
A
red
T
y
0
for the pivot in terms of the free we find
y
2
=-2
y
3
y
2
2
y
3
and
y
1
=
y
3
y
1
y
3
hence
𝒩AT={y31-21|y3∈ℝ}
𝒩
A
y3
y3
1
-2
1
(4)
The procedure is no different than that used to compute the
null space
of
AA
itself. In fact
- Definition 2:
A Basis for the Left Null Space
Suppose that
AT
A
is n-by-m with pivot indices
{
c
j
|j=1…r}
j
1
…
r
c
j
and free indices
{
c
j
|j=r+1…m}
j
r
1
…
m
c
j
.
A basis for
𝒩AT
𝒩
A
may be constructed of
m-r
m
r
vectors
z
1
z
2
…
z
m
-
r
z
1
z
2
…
z
m
-
r
where
z
k
z
k
,
and only
z
k
z
k
,
possesses a nonzero in its
c
r
+
k
c
r
+
k
component.