1
LECTURE 3.1A
Dr. Vyas
Overview: (Covers section 3.3)
•
Upper Triangular Linear Systems
•
Back Substitution
•
Examples/Demos
UPPER TRIANGULAR LINEAR SYSTEMS
1.
Definition:
An
N
N
×
matrix
[
]
i j
N
N
a
×
=
A
is called upper triangular
provided
that the elements satisfy
0,
i j
a
for all i
j
=
. The
N
N
×
matrix
[
]
i j
N
N
a
×
=
A
is
called lower triangular
provided that the elements satisfy
0,
i j
a
for all i
j
=
<
.
2.
If
A
is an upper triangular matrix, then
=
A X
B
is said to be an upper
triangular system of linear equations and has the form,
11
12
13
1
1
1
1
1
22
23
2
1
2
2
2
3
3
33
3
1
3
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
N
N
N
N
N
N
N
N
N
N
N
N
N
N
NN
a
a
a
a
a
x
b
a
a
a
a
x
b
x
b
a
a
a
x
b
a
a
x
b
a








=
KK
KK
KK
M
M
M
M
MKKKMKKKM
KK
KKK
3.
The
lower triangular system
will have nonzero terms below the diagonal of the
matrix while zeroes above the diagonal,
unlike
the case above.
4.
Theorem 3.5
: Suppose that
=
A X
B
is an upper triangular system with the form
given above. If
0
kk
a
≠
for
1, 2,
.....
,
k
N
=
, then there exists a unique solution to
the upper triangular system. The unique solution is found by first determining
/
N
N
NN
x
b
a
=
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 Spring '08
 Vyas
 Math, Determinant, Linear Systems, Characteristic polynomial, upper triangular

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