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[i.e., the closed-form solution of the unidimensional basic TLS problem (1.17)].
Theorem 91 (
g
(γ )
Zeroe)
The zeros of eq.
(
4.58
)
coincide with the squared
singular values of
[
A
;
b
]
.
Proof.
Consider
(1.2)
[respectively,
(1.3)]
as
the
SVD
of
A
(respectively,
σ
n
>
σ
n
+
1
. Since the
n
+
1 singular vectors
v
i
are eigenvectors of
[
A
;
b
]) and
[
A
;
b
]
T
[
A
;
b
], its intersection [
v
i
;−
1]
T
with the TLS hyperplane satisfies the
following set:
[
A
;
b
]
T
[
A
;
b
]
A
T
AA
T
b
b
T
Ab
T
b
v
i
−
v
i
−
v
i
−
2
i
=
=
σ
(4.61)
1
1
1
Transforming (4.61) as
T
z
−
i
z
g
2
g
=
T
U
T
b
,
z
=
V
T
v
i
(4.62)
=
σ
with
g
T
2
−
b
1
1
then
T
i
I
z
−
σ
2
2
i
g
T
z
2
=
g
and
σ
+
=
b
2
. Substituting
z
in the latter
expression by the former yields
+
g
T
T
i
I
−
1
g
=
b
2
−
σ
2
2
2
σ
(4.63)
i
Recalling that
g
=
q
[see eq. (4.59)], this equation coincides with eq. (4.58) and
shows that the zeros of
g
(γ )
are the squared singular values of [
A
;
b
].Thereare
no other zeros because the characteristic equation (4.61) from which the secular
equation originates is satisfied only by its eigenvalues. Q.E.D.
Alternative Proof.
The characteristic equation of [
A
;
b
]
T
[
A
;
b
]is
det
[
A
;
b
]
T
[
A
;
b
]
−
γ
I
n
+
1
=
det
A
T
A
−
γ
I
n
b
T
b
I
n
−
1
A
T
b
b
T
A
A
T
A
−
γ
−
−
γ
=
0
(4.64)
The second factor on the right-hand side coincides with
−
g
(γ )
in eq. (4.55).
Hence, if the squared singular values of [
A
;
b
] are distinct from the squared
singular values of
A
, the zeros of
g
(γ )
coincide with the eigenvalues of [
A
;
b
]
(i.e., the squared singular values of [
A
;
b
]
)
.
Corollary 92 (
E
TLS
Critical Points)
The zeros of eq.
(
4.58
)
coincide with the
levels of E
TLS
(
x
)
at its critical points.
Proof.
The squared singular values of [
A
;
b
] are the values of the Rayleigh
quotient
E
TLS
(
x
)
at its critical points.
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