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where
x
=
−
1
w
n
+
1
(6.61)
y
=
w
n
+
1
with the property tr
xy
T
=
1, where tr represents “trace”. Assuming that the
matrix
/
2
ζ
represents the eigenvalue matrix of the TLS problem [98] if
ζ
=
0
.
5,
eq. (4.13) represents how the eigenvalues of the GeTLS/GeMCA problem are
changed with regard to TLS when
ζ
differs from 0
.
5. It changes in a relative
way as a rank one matrix proportional to a function of
ζ
. In particular, using a
first-order Taylor approximation around
ζ
=
0
.
5, it follows that
I
n
+
1
−
4
(ζ
−
0
.
5
)
xy
T
2
ζ
∼
(6.62)
which is in agreement with the analysis in Section 6.2.1.
6.5 GeMCA SPECTRA
After adding some other inequalities, we continue with the bounds for the
GeMCA critical values for every
ζ
. From eq. (6.2):
D
−
1
Ru
Ru
=
γ
Du
⇒
=
γ
u
(6.63)
Because the matrix
D
−
1
R
is the product of two positive semidefinite matrices
for 0
<ζ <
1, it can be derived [199, Th. 7.10, p. 227] that
2
i
2
(
1
−
ζ )
≤
α
i
2
σ
≤
σ
i
2
ζ
for
ζ
≤
0
.
5
(6.64)
2
i
2
i
σ
σ
≤
α
i
≤
for
ζ
≥
0
.
5
2
ζ
2
(
1
−
ζ )
2
σ
i
being the
i
th eigenvalue of matrix
R
(eigenvalues are sorted in decreasing
order). Another possible inequality can be deduced by eq. (6.2):
2
n
≤
b
T
b
≤
σ
2
1
σ
(6.65)
+
1
because
b
T
b
is a principal submatrix of
R
. Using all the bounds seen previously,
it is possible to consider all the relative positions of the GeMCA critical values
α
i
for every possible value of the parameter
ζ
. Recall [30] that
α
1
and
α
n
+
1
represent the maximum and the minimum of
E
GeTLS
, respectively. The other
eigenvalues correspond to saddles.
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