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where
a
0. These are the tridiagonal Toeplitz matrices, these matrices
are among the few nontrivial structures that admit formulas for their eigenvalues
and eigenvectors. Matrix
B
givenby(
7.8
) is the particular case of matrix
T
where
a
=
0and
b
=
1, therefore it is known that its eigenvalues are given by (
7.10
)andthe
components of the eigenvector
c
.
k
(column vector) corresponding to
=
b
=
c
=
λ
k
are given by
(
7.11
). See [
4
].
Since the
m
eigenvalues are all different, it follows that the eigenvectors are or-
thogonal, that is to say
c
T
.
k
.
k
c
k
=
0 f
k
=
.
With the constants chosen in the elements
c
hk
the equalities
m
h
=
1
c
hk
=
1,
c
T
.
k
c
.
k
=
k
=
1
,...,
m
are satisfied, this can be proved directly or by applying Lemma
2
, which we will see
below. Therefore (
7.12
) is proved.
n
Proposition 1.
The number of elements of
F
m
satisfying
A
(
1
)=
r
and
A
(
n
)=
sis
,
given by a
(
n
)
rs
, and is equal to
m
s
2
π
k
=
1
β
k
a
(
n
)
r
+
n
−
1
sin
=(
−
1
)
k
(
1
−
2cos
β
k
)
(
r
β
k
)
sin
(
s
β
k
)
rs
where
k
π
β
k
=
1
.
(7.13)
+
m
Proof.
A proof of this result can be found in [
3
]. We can provide the following
proof; let
C
and
Λ
be the given in Lemma
1
.
CC
=
I
and
C
Λ
C
=
B
is satisfied,
therefore
M
n
B
n
−
1
n
−
1
C
=
=
C
Λ
(7.14)
is fulfilled. From the above equality it follows that
m
k
=
1
λ
a
(
n
)
n
−
1
=
c
rk
c
sk
rs
k
m
k
=
1
(
1
−
2cosβ
k
)
2
r
+
s
n
−
1
=(
−
1
)
1
sin
(
r
β
k
)
sin
(
s
β
k
)
m
+
m
s
2
π
k
=
1
β
k
n
−
r
+
1
sin
=(
−
1
)
k
(
1
−
2cos
β
k
)
(
r
β
k
)
sin
(
s
β
k
)
(7.15)
where
β
k
is given by (
7.13
), and the proof is complete.
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