Math 417 homework 6 solutions
1
Section 5.2 problem 14
~u
1
=
1
10
1
7
1
7
,
~u
2
=
1
√
2

1
0
1
0
,
~u
3
=
1
√
2
0
1
0

1
.
2
Section 5.2 problem 18
Q
=
1
5

4

3
0
0
0
1

3
4
0
,
R
=

5

5
0
0

35
0
0
0

2
.
3
Section 5.3 problem 6,8,10

B
is orthogonal as well:
(

B
)
T
(

B
) =
B
T
B
=
I.
A
+
B
is not always orthogonal. For example
A
=
I
and
B
=
I
are orthog
onal, but
A
+
B
= 2
I
is not:
(2
I
)
T
(2
I
) = 4
I
6
=
I.
B

1
AB
is orthogonal too: note that
B

1
=
B
T
for orthogonal matrices, so
(
B

1
AB
)
T
(
B

1
AB
) = (
B
T
AB
)
T
(
B
T
AB
) =
B
T
A
T
(
B
T
)
T
B
T
AB
=
B
T
A
T
BB
T
AB.
BB
T
=
I
as well, so after three cancellations we get =
I
.
4
Section 5.3 problem 28
An orthogonal transformation
L
has a matrix
A
that is orthogonal:
L
(
~v
) =
A~v.
Then
L
(
~v
)
·
L
(
~w
) = (
A~v
)
·
(
A~w
) = (
A~v
)
T
(
A~w
) =
~v
T
A
T
A~w.
1
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Here we exploit that
A
is orthogonal:
=
~v
T
~w
=
~v
·
~w.
5
Least squares problem:
We have to find
A
so that
~
f
=
A~x.
To this end, calculate the entries of
~
f
. For example
f
(

1) =
x
2
(

1)
2
+
x
1
(

1) +
x
0
=
x
0

x
1
+
x
2
.
Continuing in this way, we see that
A
=
1

1
1
1
0
0
1
1
1
1
1
1
.
The rest can be done in Matlab:
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 Fall '07
 ELLING
 Math, Matrices, mean square error, orthogonal matrices, orthogonal transformation

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