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Mathematic 104, Fall 2010: Assignment #6
Due:
Wednesday, November 17th
Instructions:
Please ensure that your answers are legible. Also make sure that all steps are shown – even
for problems consisting of a numerical answer. Bonus problems cover advanced material and, while good
practice, are
not
required and will
not
be graded.
Problem #1.
Consider the matrix
A
=
±
1
2
0
2
²
a) By hand compute the SVD of
A
. To do this it is useful to recall that all vectors in
x
with
||
x
||
2
= 1
are of the form
x
=
±
cos
θ
sin
θ
²
.
b) Using the SVD determine the rank one matrix
B
that best approximates
A
in the Frobenius norm.
c) Compare how well
B
approximates
A
in the Frobenius norm with how well the rank one matrices
A
1
=
±
1
0
0
0
²
and
A
2
=
±
0
2
0
2
²
approximate
A
in the Frobenius norm.
Problem #2.
Excercise 4.4 of Lecture 4 of Trefethen-Bau.
Problem #3.
Let
A
1
,A
2
∈
C
m
×
m
suppose that the left singular vectors of
A
1
are
{
u
1
1
,...,u
1
m
}
and the
right singular vectors are
{
v
1
1
,...,v
1
m
}
while the left singular vectors of

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