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where
√
2
ζ
I
n
0
√
2
Z
=
(6.16)
0
T
(
1
−
ζ )
Define matrix
K
as
A
T
A
ζ
A
T
b
√
ζ (
1
−
ζ )
=
Z
−
T
[
A
;
b
]
T
[
A
;
b
]
Z
−
1
1
2
K
=
Z
−
T
RZ
−
1
=
b
T
A
√
ζ (
b
T
b
1
−
ζ
1
−
ζ )
A
√
2
ζ
;
2
=
[
A
b
]
Z
−
1
b
√
2
(
1
−
ζ )
2
2
;
=
(6.17)
2
and the corresponding eigendecomposition is given by
V
T
KV
=
diag
(α
1
,
...
,
α
n
+
1
)
=
D
α
(6.18)
The eigenvector matrix
Y
is defined as
Y
=
y
1
,
...
,
y
n
+
1
=
Z
−
1
V
(6.19)
and is
D
-orthogonal; that is,
Y
T
DY
=
I
(6.20)
Hence,
(α
i
,
y
i
)
is an eigenpair of the symmetric positive definite pencil
(
R
,
D
)
.
From a MCA (minor components analysis) point of view, the theory above
can be reformulated as a Rayleigh quotient:
T
u
T
Ru
u
T
Du
=
u
T
Ru
u
T
Z
T
Zu
1
2
(
Ax
−
b
)
(
Ax
−
b
)
E
GeTLS
(
x
)
=
=
(
1
−
ζ )
+
ζ
x
T
x
T
Z
−
T
RZ
−
1
T
K
v
v
=
v
v
=
v
=
E
MCA
(v)
(6.21)
v
T
v
T
v
where
u
=
Z
−
1
v
(6.22)
c
of
E
MCA
correspond to the columns of matrix
V
defined
in eq. (6.18). Hence, eq. (6.22) corresponds to eq. (6.19). The associated eigen-
values
α
i
are the corresponding values of
E
GeTLS
From a theoretical point of
view, this equivalence GeTLS MCA has very important implications; the MCA
theory [30] can be used for explaining the GeTLS behavior for all values of the
parameter
ζ
. From a numerical point of view, it means that like the MCA EXIN
algorithm (neuron) [30] (basically, a gradient flow of the Rayleigh quotient),
The critical vectors
v
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