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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 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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