Geology Reference
In-Depth Information
Since
j
j
c
−
1
jk
c
kj
=
δ
=
N
+
1,
(2.430)
the entropy becomes
1)
logπ
−
1
.
H
=
log
|
C
|+
(
N
+
2log
a
+
(2.431)
Choosing the constant
a
=
√
π
e
, the final expression for the entropy is
H
=
log
|
C
|=
log (det
T
N
).
(2.432)
1.
For processes of indefinite duration we use the
entropy density
or
entropy rate
which Middleton (1960, p. 306) defined as
Thus far, we have considered the entropy of processes with finite length
N
+
log
(det
T
N
)
1/(
N
+
1)
.
H
h
=
lim
N
→∞
1
=
lim
N
→∞
(2.433)
+
N
The Toeplitz matrix
T
N
has non-negative definite eigenvalues. For eigenvalue λ
and corresponding eigenvector
x
,bydefinition,
T
N
x
=
λ
x
.
(2.434)
Left multiplication gives
x
H
T
N
x
=
λ
x
H
x
,
(2.435)
where the superscript
H
indicates the adjoint. Then,
k
=
0
φ(
j
N
N
x
H
T
N
x
k
)
x
∗
j
x
k
=
−
j
=
0
N
N
E
z
j
z
∗
k
x
∗
j
x
k
=
j
=
0
k
=
0
⎩
⎭
N
N
z
j
x
∗
j
·
z
∗
k
x
k
=
E
j
=
0
k
=
0
⎩
2
⎭
≥
z
j
x
∗
j
N
=
E
0.
(2.436)
j
=
0
Thus, as stated,
x
H
T
N
x
x
H
x
≥
λ
=
0.
(2.437)
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