Consider the ( n × n ) symmetric matrix A = ( a i j ) defined as follows: a) a i j = 1 , 1 ≤ i ≤ n ; . b) a i j = − 1 , i ≠ j , 1 ≤ i , j ≤ n . ( A ( 4 × 4 ) version of this matrix is given in Example 7.) Verify that the eigenvalues of A are λ = 2 (geometric multiplicity n − 1 ) and λ = 2 − n (geometric multiplicity 1 ). [ Hint: Show that the following are eigenvectors: u i = e 1 − e i , 2 ≤ i ≤ n and u i = [ 1 , 1 , .... , 1 ] T .]
Consider the ( n × n ) symmetric matrix A = ( a i j ) defined as follows: a) a i j = 1 , 1 ≤ i ≤ n ; . b) a i j = − 1 , i ≠ j , 1 ≤ i , j ≤ n . ( A ( 4 × 4 ) version of this matrix is given in Example 7.) Verify that the eigenvalues of A are λ = 2 (geometric multiplicity n − 1 ) and λ = 2 − n (geometric multiplicity 1 ). [ Hint: Show that the following are eigenvectors: u i = e 1 − e i , 2 ≤ i ≤ n and u i = [ 1 , 1 , .... , 1 ] T .]
Solution Summary: The author explains the symmetric matrix A=(a_ij) defined as follows.
Consider the
(
n
×
n
)
symmetric matrix
A
=
(
a
i
j
)
defined as follows:
a)
a
i
j
=
1
,
1
≤
i
≤
n
;
.
b)
a
i
j
=
−
1
,
i
≠
j
,
1
≤
i
,
j
≤
n
.
( A
(
4
×
4
)
version of this matrix is given in Example 7.) Verify that the eigenvalues of
A
are
λ
=
2
(geometric multiplicity
n
−
1
) and
λ
=
2
−
n
(geometric multiplicity
1
). [Hint: Show that the following are eigenvectors:
u
i
=
e
1
−
e
i
,
2
≤
i
≤
n
and
u
i
=
[
1
,
1
,
....
,
1
]
T
.]
Definition Definition Matrix whose transpose is equal to itself. For a symmetric matrix A, A=AT.
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