Solutions for Homework 2 Foundations of Computational
Math 1 Fall 2010
Problem 2.1
Let
n
= 4 and consider the lower triangular system
Lx
=
f
of the form
1
0
0
0
λ
21
1
0
0
λ
31
λ
32
1
0
λ
41
λ
42
λ
43
1
ξ
1
ξ
2
ξ
3
ξ
4
=
φ
1
φ
2
φ
3
φ
4
Recall, that it was shown in class that the columnoriented algorithm could be derived
from a factorization
L
=
L
1
L
2
L
3
where
L
i
was an elementary unit lower triangular matrix
associated with the
i
th column of
L
.
Show that the roworiented algorithm can be derived from a factorization of
L
of the
form
L
=
R
2
R
3
R
4
where
R
i
is associated with the
i
th row of
L
.
Solution:
We can deﬁne
R
i
in a manner similar to the column forms used for
L
i
. Speciﬁcally,
deﬁne for the case
n
= 4
R
2
=
1
0 0 0
λ
21
1 0 0
0
0 1 0
0
0 0 1
R
3
=
1
0
0 0
0
1
0 0
λ
31
λ
32
1 0
0
0
0 1
R
4
=
1
0
0
0
0
1
0
0
0
0
1
0
λ
41
λ
42
λ
43
1
It is straighforward to verify that this satisﬁes
L
=
R
2
R
3
R
4
. To see that the pattern holds
for any
n
note that
R
j
=
I
+
e
j
r
T
j
r
T
j
e
k
= 0
, j
≤
k
≤
n
R
i
R
j
=
R
i
(
I
+
e
j
r
T
j
) =
R
i
+
R
i
e
j
r
T
j
=
R
i
+
e
j
r
T
j
,
since
i < j
1
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View Full Documentand therefore we simply put the nonzeros of
j
th row from
R
j
into the zero positions in the
j
th row of
R
i
to form the product. This easily generalizes when
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 Spring '11
 Gallivan
 Determinant, Characteristic polynomial, Triangular matrix, Rj ej

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