Homework 3 Numerical Linear Algebra 1 Fall 2010
Due date: beginning of class Monday, 10/4/10
Problem 3.1
Problem 3.1.6 Golub and Van Loan p. 93
Problem 3.2
Recall that any unit lower triangular matrix
L
∈
n
×
n
can be written in factored form as
L
=
M
1
M
2
· · ·
M
n

1
(1)
where
M
i
=
I
+
l
i
e
T
i
is an elementary unit lower triangular matrix (column form). Given
the ordering of the elementary matrices, this factorization did not require any computation.
Consider a simpler elementary unit lower triangular matrix (element form) that differs
from the identity in
one offdiagonal element
in the strict lower triangular part, i.e.,
E
ij
=
I
+
λ
ij
e
i
e
T
j
where
i
=
j
.
3.2.a
. Show that computing the product of two element form elementary matrices is
simply superposition of the elements into the product given by
E
ij
E
rs
=
I
+
λ
ij
e
i
e
T
j
+
λ
rs
e
r
e
T
s
whenever
j
=
r
.
3.2.b
. Show that if
j
=
r
and
i
=
s
then computing
E
ij
E
rs
with requires no compu
tation and
E
ij
E
rs
=
E
rs
E
ij
i.e., the matrices commute.
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 Fall '06
 gallivan
 Ri, Eij Ers, λij ei

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