Homework Assignment #6
16.2 Let
Q
(
x
) =
x
T
A
x
be a quadratic form on
R
n
. By evaluating
Q
on each of the coordinate axes in
R
n
,
prove that a necessary condition for a symmetric matrix to be positive definite (positive semidefinite)
is that all the diagonal entries be positive (nonnegative). State and prove the corresponding result for
negative and negative semidefinite matrices. Give an example to show that this necessary condition is
not sufficient.
Answer:
The corresponding result is that a necessary condition for a symmetric matrix to be negative
definite (negative semidefinite) is that all the diagonal entries be negative (nonpositive).
All can be proven together by considering
Q
(
e
i
) =
e
T
i
A
e
i
=
a
ii
where
e
i
is the
i
t
h
standard basis
vector. The sign of this is determined by the type of definiteness or semidefiniteness. If
A
is positive
definite, then
a
ii
>
0 for all
i
= 1
, . . . , n
. If
A
is positive semidefinite, then
a
ii
≥
0 for all
i
= 1
, . . . , n
.
If
A
is negative definite, then
a
ii
<
0 for all
i
= 1
, . . . , n
. If
A
is negative semidefinite, then
a
ii
≤
0 for
all
i
= 1
, . . . , n
.
As for examples, consider
A
1

1
4
4

1
and
A
2
0
4
4
0
.
Then
A
!
satisifes the necessary conditions for negative definiteness, and
A
2
satisfies the necessary condi
tions for negative semidefiniteness. But
Q
1
(1
,
1) = 6 and
Q
2
(1
,
1) = 8, showing that
A
1
is not negative
definite and that
A
2
is not negative semidefinite.
16.6 Determine the definiteness of the following constrained quadratics.
a
)
Q
(
x
1
, x
2
) =
x
2
1
+ 2
x
1
x
2

x
2
2
, subject to
x
1
+
x
2
= 0.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 STAFF
 Optimization, $4

Click to edit the document details