Due: October 2, 2008
CS 257 (Luke Olson): Homework #5
Problem 1
Problem 1
a. If A is a nonsingular square matrix, prove that
A
T
A
is symmetric and positive deﬁnite. You can
assume that all values in A are real.
b. Experiment with the code below. What can you hypothesize about the condition number of
A
T
A
compared to
A
?
Listing 1: Code for problem 1b
1
n=5;
2
3
A =
rand
(n,n);
4
B = A’
*
A;
5
fprintf
(
’cond(A) cond(A)ˆ2 cond(AˆT
*
A)\n’
)
6
fprintf
(
’%g %g %g\n’
,
cond
(A),
cond
(A)ˆ2,
cond
(B))
c. Suppose you’re working on an embedded platform with no prebuilt numerical libraries, and you
need to solve
Ax
=
b
. However, you also have free access to
A
T
A
and
A
T
b
because they were
computed previously for some unrelated reason in your code. You now have a choice of writing
Cholesky factorization to solve
A
T
Ax
=
A
T
b
or Gaussian Elimination with scaled partial pivoting to
solve
Ax
=
b
.
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 Spring '08
 Olson
 Luke Olson

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