Graphics Programs Reference
In-Depth Information
EXAMPLE A4
Letting
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
123
121
0 12
1
6
8
0
A
=
u
=
v
=
−
2
−
3
v
,
Av
and
u
T
Av
.
compute
u
+
v
,
u
·
Solution
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
1
+
8
9
6
u
+
v
=
6
+
0
−
2
−
3
−
5
u
·
v
=
1(8))
+
6(0)
+
(
−
2)(
−
3)
=
14
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
a
1
·
v
a
2
·
1(8)
+
2(0)
+
3(
−
3)
−
1
5
Av
=
v
a
3
·
1(8)
+
2(0)
+
1(
−
3)
+
+
−
−
v
0(8)
1(0)
2(
3)
6
u
T
Av
=
·
=
−
+
+
−
−
=
u
(
Av
)
1(
1)
6(5)
(
2)(
6)
41
EXAMPLE A5
Compute
|
A
|
, where
A
is given in Example A4. Is
A
positive definite?
Solution
Laplace's development of the determinanton the first rowyields
1
−
2
+
3
21
12
11
0 2
12
0 1
|
A
| =
=
1(3)
−
2(2)
+
3(1)
=
2
Developmenton the third rowissomewhateasierdue to the presence of the zero
element:
0
−
1
+
2
23
21
13
11
12
12
|
A
| =
=
0(
−
4)
−
1(
−
2)
+
2(0)
=
2
To verifypositive definiteness, weevaluate the determinants of the leading
minors:
A
11
=
1
>
0O.K.
=
=
A
11
A
12
A
21
A
22
12
12
0
Not O.K.
A
is not positive definite.
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