Let B = Q T A Q , where q and A are as in Exercise 38. Consider the ( 2 × 2 ) submatrix of B given by A 1 in Exercise 38. Show that the eigenvalues of A 1 are real. [ Hint: Calculate det ( B − t I ) , and show that every eigenvalues of A 1 is an eigenvalue of B . Then make a statement showing that all the eigenvalues of B are real.] Let A be a real ( 3 × 3 ) matrix with only real eigenvalue. Suppose that A u = λ u , where u T u = 1 . By the Gram-Schmidt process, there are vectors v and w in R 3 such that { u , v , w } is an orthonormal set. Consider the orthogonal matrix Q given by Q = [ u , v , w ] . Verify that Q T A Q = [ λ u T A v u T A w 0 v T A v v T A w 0 w T A v w T A w ] = [ λ u T A v u T A w 0 A 1 0 ]
Let B = Q T A Q , where q and A are as in Exercise 38. Consider the ( 2 × 2 ) submatrix of B given by A 1 in Exercise 38. Show that the eigenvalues of A 1 are real. [ Hint: Calculate det ( B − t I ) , and show that every eigenvalues of A 1 is an eigenvalue of B . Then make a statement showing that all the eigenvalues of B are real.] Let A be a real ( 3 × 3 ) matrix with only real eigenvalue. Suppose that A u = λ u , where u T u = 1 . By the Gram-Schmidt process, there are vectors v and w in R 3 such that { u , v , w } is an orthonormal set. Consider the orthogonal matrix Q given by Q = [ u , v , w ] . Verify that Q T A Q = [ λ u T A v u T A w 0 v T A v v T A w 0 w T A v w T A w ] = [ λ u T A v u T A w 0 A 1 0 ]
Solution Summary: The author explains that the eigenvalues of A_1 are real.
Let
B
=
Q
T
A
Q
, where
q
and
A
are as in Exercise 38. Consider the
(
2
×
2
)
submatrix of
B
given by
A
1
in Exercise 38. Show that the eigenvalues of
A
1
are real. [Hint: Calculate det
(
B
−
t
I
)
, and show that every eigenvalues of
A
1
is an eigenvalue of
B
. Then make a statement showing that all the eigenvalues of
B
are real.]
Let
A
be a real
(
3
×
3
)
matrix with only real eigenvalue. Suppose that
A
u
=
λ
u
, where
u
T
u
=
1
. By the Gram-Schmidt process, there are vectors
v
and
w
in
R
3
such that
{
u
,
v
,
w
}
is an orthonormal set. Consider the orthogonal matrix
Q
given by
Q
=
[
u
,
v
,
w
]
. Verify that
Q
T
A
Q
=
[
λ
u
T
A
v
u
T
A
w
0
v
T
A
v
v
T
A
w
0
w
T
A
v
w
T
A
w
]
=
[
λ
u
T
A
v
u
T
A
w
0
A
1
0
]
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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