Department of Mathematics
University of Houston
Numerical Analysis I
Dr. Ronald H.W. Hoppe
Numerical Mathematics I
(1. Homework Assignment)
Exercise 1
(
Gauss elimination for diagonally dominant matrices
)
A matrix
A
∈
R
n
×
n
is called diagonally dominant, if

a
ii
 ≥
X
j
6
=
i

a
ij

,
1
≤
i
≤
n .
Let
A
(
k
)
,
0
≤
k
≤
n

1, with
A
(0)
:=
A
be the intermediate matrices that arise
during Gauss elimination.
Show by an induction argument that the matrices
A
(
k
)
,
1
≤
k
≤
n

1, are also
diagonally dominant and satisfy
n
X
j
=
k
+1

a
(
k
)
ij
 ≤
n
X
j
=1

a
ij

.
Show further that this implies

a
(
k
)
ij
 ≤
2 max
i,j

a
ij

.
Exercise 2
(
Avoiding fillIn
)
Let
A
∈
R
5
×
5
be a regular matrix of the following form
A
=
×
×
×
×
×
×
×
×
×
×
×
×
×
.
Here, the symbol
×
denotes a nonzero entry, whereas all other matrix entries are
supposed to be zero.
(i) Show that in case of Gauss elimination without pivoting, the first step already
generates nonzero entries at all places where the original entry was zero (”fillin”).
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 Spring '12
 Staff
 Matrices, gauss elimination, ∈ Rn, diagonally dominant matrices

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