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⎡
1100
...
0
⎤
⎣
⎦
1110
...
0
B
=
0111
0
.... . .
0000
...
,
(7.8)
...
1
that is the matrix having zeros as elements everywhere, with the exception of the
elements of the principal diagonal and that immediately above and below which are
equal to 1.
If we call
M
(
n
)
the square matrix of order
m
whose elements are the
a
(
n
)
rs
,
r
=
1
,...,
m
,
s
=
1
,...,
m
a
(
n
)
rs
M
(
n
)
=
,
(7.9)
it is clear that
M
(
2
)
=
,
B
M
(
n
)
=
M
(
n
−
1
)
B
,
and so
a
(
n
)
rs
M
(
n
)
=
B
n
−
1
=
.
The following lemma and proposition are known, they give us the tools to obtain
the cardinalities of the sets
n
n
m
.
Lemma 1.
The eigenvalues of the square matrix B of order m, given by (
7.8
)are
F
m
and
F
,
,
k
π
λ
k
=
−
1
,
=
,...,
,
1
2cos
k
1
m
(7.10)
m
+
and the components of the eigenvector c
.
k
(column vector) corresponding to
λ
k
are
k
2
m
1
sin
hk
π
h
+
c
hk
=(
−
1
)
,
h
=
1
,...,
m
.
(7.11)
+
+
m
1
Let us denote by C the square matrix of order m whose elements are the c
hk
,C
=
[
c
hk
]
. Then C is an orthogonal and symmetric matrix, that is to say,
C
T
=
C
,
CC
=
I
,
C
Λ
C
=
B
(7.12)
is satisfied, where
Λ
is the diagonal matrix whose elements of the principal diagonal
are the eigenvalues
λ
k
.
Proof.
Let
T
be the square matrix
⎡
⎣
⎤
⎦
bc
00
···
0
ab c
0
···
0
0
abc
0
......
00
···
T
=
,
···
abc
00
···
0
ab
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