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)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '10
 WILKIE
 Math, Matrices, Eigenvalues, Diagonal matrix, Orthogonal matrix, QP T

Click to edit the document details