COT4501 Spring 2012
Homework III
This assignment has six problems and they are equally weighted. The assignment is due in
class on Tuesday, February 21, 2012. There are four regular problems and two computer prob
lems (using MATLAB). For the computer problems, turn in your results (e.g., graphs, plots,
simple analysis and so on) and also a printout of your (MATLAB) code.
Problem 1
1. Show that if the vector
v
6
= 0
, then the matrix
H
=
I

2
vv
>
v
>
v
is orthogonal and symmetric.
2. Let
a
be any nonzero vector. If
v
=
a

α
e
1
, where
α
=
± k
a
k
2
, and
H
=
I

2
vv
>
v
>
v
,
show that
Ha
=
α
e
1
.
Problem 2
Consider the vector
a
as an
n
×
1
matrix.
1. Write out its QR factorization, showing the matrices
Q
and
R
explicitly.
2. What is the solution to the linear least squares problem
a
x
’
b
, where
b
is a given
n
vector.
3. How do you interpret the above result geometrically?
Problem 3
Consider the following matrix
A
A
=
1
1
0
0
,
where
is a positive number smaller than
√
mach
in a given floatingpoint system. In class,
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 Spring '08
 Davis
 Singular value decomposition, Orthogonal matrix, Linear least squares, QR algorithm, Householder transformation

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