Spring 2010
CS530
– Analysis of Algorithms
Homework 2
Homework 2, due Feb 3
You must prove your answer to every question.
Problems with a (
*
) in place of a score may be a little too advanced, or too challenging to
most students, so I do not assign a score to them. But I will still note if you solve them.
Problem 1
(28.16)
.
(10pts) Let
A
,
B
be
n
×
n
matrices such that
AB
=
I
. Prove that if
A
is obtained from
A
by adding row
j
into row
i
=
j
then the inverse
B
of
A
is obtained by
subtracting column
i
from column
j
of
B
.
Solution.
Let
T
=
(
t
kl
) be the matrix obtained from the unit matrix
I
after we add row
j
into row
i
. Thus,
t
kl
=
1 if
k
=
l
or if
k
=
i
,
l
=
j
. It is clearly the same matrix as the one we
obtain from
I
by adding column
j
to column
i
. It is easy to check that the inverse
T

1
is
obtained if we subtract row
j
of
I
from row
i
, and that this is the same as when we work on
the columns of
I
instead of rows.
It is easy to check that
A
=
TA
and hence
B
=
BT

1
. But
BT

1
is obtained by subtract
ing column
j
of
B
from column
i
.
Problem 2.
(10pts) In class, I have shown how to compute the Fourier transform fast,
writiting up a recurrence
F
(
n
)
2
·
F
(
n
/2)
+
c
·
n
for the cost of transforming a vector of length
n
, (here
c
is a constant). I indicated that this leads to the inequality
F
(
n
)
d
·
n
log
n
for
some constant
d
. Derive this inequality now.
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 Spring '09
 Linear Algebra, Algorithms, Vector Space, Analysis of algorithms, Row, row vectors

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