Information Technology Reference
In-Depth Information
s1
min
s1
max
min
Figure 5.2
TLS equilevel curves for a benchmark overdetermined system with the corre-
sponding critical points and the barrier of convergence, together with a section along the
straight line passing through the minimum and the saddle.
the following deductions are found, assuming
A
T
A
nonsingular:
⇒
v
j
T
=
λ
j
v
j
A
T
A
−
1
=
λ
j
v
j
A
T
A
−
1
A
T
v
j
A
T
A
=
λ
j
v
T
⇒
v
j
A
T
j
Recalling the definition of pseudoinverse of a matrix, we have
v
j
A
T
=
λ
j
v
j
A
+
(5.60)
Hence, from the definition of
q
j
,
v
j
A
T
b
=
λ
j
v
j
TA
+
b
=
λ
j
A
+
b
·
v
j
q
j
=
(5.61)
where
·
denotes a scalar product. Recalling that
x
OLS
=
A
+
b
, it follows that
=
λ
j
x
·
v
j
q
j
(5.62)
that is, the convergence keys are the components of the OLS solution
x
=
A
+
b
with respect to the right singular vectors
v
j
of
A
, weighted by the corresponding
squared singular values
λ
j
.
Theorem 102 (Second Characterization)
The convergence keys are the com-
ponents of A
T
b with respect to the unidimensional eigenspaces of A
T
A.
Search WWH ::
Custom Search