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A
;
B
]
=
is called a
TLS solution (
the corresponding TLS correction is
[
[
A
;
B
]
−
[
A
;
B
]
)
.
Theorem 17 (Solution of the Multidimensional TLS Problem)
Given
(
1.3
)
as the SVD of
[
A
;
B
]
,if
σ
n
>
σ
n
+
1
,then
=
U
1
1
V
11
;
V
21
[
A
;
B
]
=
U
diag
(σ
1
,
...
,
σ
n
,0,
...
,0
)
V
T
(1.26)
with corresponding TLS correction matrix
=
U
2
2
V
12
;
V
22
A
;
B
]
[
(1.27)
solves the TLS problem
(
1.24
)
and
X
V
12
V
−
1
22
=−
(1.28)
exists and is the unique solution to A X
=
B.
Proof.
See [98, p. 52].
Theorem 18 (Closed-Form Multidimensional TLS Solution)
Given
(
1.2
)
[
respectively,
(
1.3
)]
as the SVD of A
(
[
A
;
B
]
)
,if
σ
n
>
σ
n
+
1
=···=
σ
n
+
d
,then
X
=
A
T
A
−
σ
1
I
−
1
A
T
B
2
n
(1.29)
+
Proof.
See [98, Th. 3.10].
Proposition 19 (Existence and Uniqueness Condition)
See
[
98, p. 53
]
:
σ
n
> σ
n
+
1
⇒
σ
n
> σ
n
+
1
and
V
22
is nonsingular.
Remark 20
The multidimensional problem AX
B
m
×
d
can also be solved by
computing the TLS solution of each subproblem Ax
i
≈
,
d , sepa-
rately. The multidimensional TLS solution is better at least when all data are
equally perturbed and all subproblems Ax
i
≈
b
i
,
i
=
1,
...
≈
b
i
have the same degree of incom-
patibility.
1.6.2 Nonunique Solution
Theorem 21 (Closed-Form Minimum Norm TLS Solution)
Given the TLS
problem
(
1.24
)
and
(
1.25
)
, assuming that
σ
p
> σ
p
+
1
=···=
σ
n
+
d
with p
≤
n,
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