Example 6.
Suppose that we have a production function Q
=
kx
a
y
b
. Then,
∂
Q
∂
x
=
akx
a

1
y
b
,
∂
Q
∂
y
=
bkx
a
y
b

1
∂
2
Q
∂
x
∂
y
=
abkx
a

1
y
b

1
,
∂
2
Q
∂
y
∂
x
=
abkx
a

1
y
b

1
14 Some Linear Algebra
14.1 Definiteness of Quadratic Forms (
§
16.2)
Definition 27.
Let A be an m
×
m symmetric matrix, then A is
(a)
positive definite
if
x
T
A
x
>
0 for all
x
negationslash
=
0
in
R
m
,
(b)
positive semidefinite
if
x
T
A
x
≥
0 for all
x
in
R
m
,
(c)
negative definite
if
x
T
A
x
<
0 for all
x
negationslash
=
0
in
R
m
,
(d)
negative semidefinite
if
x
T
A
x
≤
0 for all
x
in
R
m
,
(e)
indefinite
if
x
T
A
x
>
0 for some
x
in
R
m
and
<
0
for some other
x
in
R
m
.
•
Q
(
x
,
y
) =

3
x
2

6
y
2
≤
0 and strictly negative except at
0
. Hence, it is negative
definite, and
(
x
,
y
) =
0
is its global maximizer.
A
=
?
•
Q
3
(
x
,
y
) =
3
x
2

6
y
2
is indefinite.
A
=
?
•
Q
4
(
x
,
y
) =
x
2
+
2
xy
+
y
2
is positive semidefinite.
A
=
?
•
Q
5
(
x
,
y
) =

(
x

y
)
2
is negative semidefinite.
A
=
?
14.2 Symmetric Matrices (
§
23.7)
Definition 28
(p.579, 582)
.
Let A be a m
×
m matrix,
v
negationslash
=
0
be a mvector and r be a
number. If A
v
=
r
v
, then
v
is called an
eigenvector
(or
characteristic vector
) of A, and
r is called an
eigenvalue
(or
characteristic value
) of A.
Example 7.
Consider A
=
parenleftbigg
2
4
4
2
parenrightbigg
,
v
=
parenleftbigg
1

1
parenrightbigg
,
w
=
parenleftbigg
1
1
parenrightbigg
. Then,
A
v
=
parenleftbigg

2
2
parenrightbigg
=

2
v
, and A
w
=
parenleftbigg
6
6
parenrightbigg
=
6
w
. Hence, the eigenvalues of A are 2
and 6.
Theorem 44
(Thm 23.16, p.621)
.
Let A be a symmetric
m
×
m matrix. Then there is a
nonsingular matrix P whose columns
w
1
,
···
,
w
m
are eigenvectors of A such that
(i)
w
1
,
···
,
w
m
are mutually orthogonal to each other
40