Assignment 2
Due : Wednesday January 27, 2010
1. The matrix factor
L
that emerges from Gaussian elimination with partial pivoting is almost
always surprisingly well conditioned. The reason for this is not fully understood.
(a) Explain why all the entries of
L
below the diagonal have absolute value at most 1, provided
partial pivoting is used.
(b) Generate random lower triangular matrices with 1’s on the diagonal and uniformly dis
tributed random numbers in the interval [
−
1
,
1] below the diagonal. (Forming a matrix of
this kind can be done using the
rand
,
tril
, and
eye
functions in Matlab. Since
rand
gen
erates random numbers uniformly distributed in [0
,
1], you should use
2*rand(n,n)1
to
obtain the requested distribution.) Make a plot of condition number versus matrix size
n
.
Use a logarithmic scale for the yaxis (condition number). The functions
cond
and
semilogy
will be helpful. Use values of
n
up to
n
= 200 and explain the apparent ﬂattening (within
ﬂuctuations) of the plot.
(c) Generate random lower triangular matrices of the same structure using the following
approach: form a random matrix
A
(using
rand
) and then extract its
L
factor from Gaussian
elimination by invoking either the builtin function
lu
or the function
lutx
from Moler’s
book. Make the same plot for this distribution as in part (b).
(d) Hand in: listings of your mﬁles and the two requested plots. In addition, describe
qualitatively the diﬀerence between the two plots, which should be immediately apparent.
2. Let
L
be an
n
×
n
lower triangular matrix with all nonzero entries on the diagonal.
(a) Explain why the following inequality holds: for all
i
= 1
, . . . , n
,
∥
L
∥
∞
≥ 
L
(
i, i
)

.
(b) Give a formula for the (
i, i
) entry of
L
−
1
in terms of the entries of
L
. Explain your
formula by considering the equation
Lx
=
e
i
, where
x
is the
i
th column of
L
−
1
and
e
i
is the
i
th column of
I
.
(c) Combine (a) and (b) to derive the following inequality:
κ
∞
(
L
)
≥
max
i
=1
,...,n

L
(
i, i
)

min
i
=1
,...,n

L
(
i, i
)

.
3. It can be shown that the deﬁnitions of the 1, 2, or
∞
norms of a vector
x
and of a matrix
A
given in lecture have the property that
∥
A
∥
= max
∥
Ax
∥
∥
x
∥
:
x
̸
= 0
,
which alternatively can be used to deﬁne the matrix norm in terms of the corresponding
vector norm. The norm is then called an “induced” or “operator” norm. (The Frobenius
norm is not an induced norm.)
(a) Let
m
= min
{∥
Ax
∥
/
∥
x
∥
:
x
̸
= 0
}
. Assuming
m
̸
= 0, show that
m
= 1
/
∥
A
−
1
∥
.
Use the
operator norm deﬁnition so that your result applies to all operator norms. [Hint: apply a
carefully chosen change of variables to
x
.]
(b) Establish the following identity, which holds for all invertible matrices
A
and all operator
norms:
∥
A
∥∥
A