Geology Reference
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If we form the matrix
X
with the eigenvectors of
T
N
as columns, then
T
N
X
=
DX
(2.438)
with
⎝
⎠
λ
0
0
λ
1
D
=
(2.439)
.
.
.
0
λ
N
the diagonal matrix of eigenvalues. On taking determinants of both sides of (2.438),
we have
det
T
N
·
det
X
=
det
D
·
det
X
(2.440)
or
det
T
N
=
det
D
=
λ
0
λ
1
···
λ
N
≥
0.
(2.441)
Hence,
T
N
is non-negative definite.
The determinant of
T
N
can be related to the spectral density
S
(
f
) through a
remarkable theorem of G. Szego (1920), (Widom, 1965, p. 202), which states that
for
F
a continuous function,
f
N
+
F
(λ
1
)
+···+
F
(λ
N
)
F
2
f
N
S
(
f
)
df
,
F
(λ
0
)
1
2
f
N
lim
N
→∞
=
(2.442)
N
+
1
−
f
N
with
f
N
denoting the Nyquist frequency. Taking
F
to be the logarithmic function,
with substitution from (2.441), we have
log
(λ
0
λ
1
···
λ
N
)
1/(
N
+
1)
logλ
0
+
logλ
1
+···+
logλ
N
lim
N
=
lim
N
N
+
1
→∞
→∞
log
(
det
T
N
)
1/(
N
+
1)
=
lim
N
→∞
f
N
log
2
f
N
S
(
f
)
df
.
1
2
f
N
=
(2.443)
−
f
N
Expanding the right-hand side produces
f
N
log
S
(
f
)
df
f
N
1
2
f
N
log
(
2
f
N
)
df
+
−
f
N
−
f
N
f
N
log
S
(
f
)
df
.
1
2
f
N
=
log (2
f
N
)
+
(2.444)
−
f
N
Taking exponentials produces
2
f
N
exp
1
2
f
N
f
N
log
S
(
f
)
df
(det
T
N
)
1/(
N
+
1)
lim
N
→∞
=
.
(2.445)
−
f
N
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