Math 54 Quiz 8
Mike Hartglass
November 6, 2010
1.) Consider the matrix
A
=
±
1 3
3 1
²
. Find matrices
Q
and
D
where
Q
is an
or
thogonal
matrix,
D
is a diagonal matrix and
A
=
QDQ

1
.
The characteristic polynomial is (
t

1)
2

9 so its roots are

2 and 4. For
t
= 2,
A

tI
=
±
3 3
3 3
²
which is easily seen to have null space generated by (1
,

1)
T
. Similarly,
the null space for
A

4
I
is generated by the vector (1
,
1)
T
. Normalizing these vectors
(as we require
Q
to be an orthogonal matrix) we see that
Q
=
±
1
√
2
1
/
√
2
1
/
√
2

1
√
2
²
and
D
=
±
4 0
0 2
²
.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2.) Let
V
be the space
C
[

1
,
1] with inner product
h
f,g
i
=
´
1

1
f
(
x
)
g
(
x
)
dx
. Find an
orthogonal basis for the subspace spanned by the polynomials 1,
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 yildiz
 Math, Linear Algebra, Matrices, Gram Schmidt, 21 gram, t2 dt, Mike Hartglass

Click to edit the document details