Math 235
Assignment 6 Solutions
1.
Show that the following are equivalent for a symmetric matrix
A
:
(1)
A
is orthogonal
(2)
A
2
=
I
(3) All the eigenvalues of
A
are
±
1
Solution: (1)
⇒
(2)
If
A
is orthogonal then
I
=
AA
T
=
AA
, since
A
is symmetric.
(2)
⇒
(3)
Assume that
A
2
=
I
and
λ
is an eigenvalue of
A
with eigenvector
~v
. Then
A~v
=
λ~v
A
2
~v
=
A
(
λ~v
)
~v
=
λA~v
~v
=
λ
2
~v
Hence
λ
2
= 1, so
λ
=
±
1.
(3)
⇒
(1) If all the eigenvalues of
A
are
±
1, then since
A
is symmetric there exists an
orthogonal matrix
P
such that
P
T
AP
=
D
where the diagonal entries of
D
are the eigen
values of
A
. Hence since the eigenvalues of
A
are
±
1 and
D
is diagonal, we have
DD
T
=
I
and so
I
=
DD
T
= (
P
T
AP
)(
P
T
AP
)
T
= (
P
T
AP
)(
P
T
A
T
P
) =
P
T
AA
T
P
⇒
I
=
A
T
A.
Thus
A
is orthogonal.
2.
Show that the following are equivalent for two symmetric matrices
A
and
B
(1)
A
and
B
are similar
(2)
A
and
B
have the same eigenvalues
(3)
A
and
B
are orthogonally similar
Solution: (1)
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 Spring '10
 WILKIE
 Math, Matrices, Eigenvalues, Diagonal matrix, Orthogonal matrix, QP T

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