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Proof.
It follows directly from the definition of
q
:
=
v
j
A
T
b
=
A
T
b
·
v
j
q
=
V
T
A
T
b
⇒
q
j
∀
j
=
1,
...
,
n
(5.63)
·
where
denotes a scalar product.
Corollary 103 (Characterization Equivalence)
The two convergence key
characterizations are equivalent.
Proof.
Using the SVD of
A
, the pseudoinverse
A
+
can be expressed as
1
σ
j
v
j
u
T
A
+
=
(5.64)
j
j
Hence, the OLS solution can be expressed as
x
=
A
+
b
=
j
1
σ
j
v
j
u
j
b
(5.65)
From eq. (5.62),
=
λ
j
k
σ
k
v
k
u
k
b
·
v
j
=
λ
j
x
·
v
j
1
1
σ
j
u
j
b
=
σ
j
u
j
b
q
j
=
λ
j
(5.66)
From the definition of
q
,
=
v
j
A
T
b
=
σ
j
u
j
b
q
j
(5.67)
Then eq. (5.66) is equivalent to eq. (5.67).
Proposition 104 (Convergence Key Property)
The following property for the
convergence keys holds:
n
q
i
λ
i
b
T
b
≥
(5.68)
i
=
1
=
σ
2
i
(λ
i
)
, where the sign
>
is valid for overdetermined systems.
Proof.
Recalling eq. (5.67), the expression on the right-hand side of the inequal-
ity (5.68) becomes
σ
i
u
i
b
2
σ
2
i
u
i
b
2
n
n
n
q
i
λ
i
=
=
(5.69)
i
=
1
i
=
1
i
=
1
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