1
6. Extrema of real valued functions
6.1 Quadratic forms (B24 Textbook Chapter 8)
A
quadratic form
is a degree 2 homogeneous polynomial function.
1
2
, ,
1
( )
(
,
,
...,
)
n
n
ij
i
j
i
j i j
f
f x
x
x
u x x
≤
=
=
=
∑
x
,
where not all
u
ij
are zero.
f(x) can be written as
[
]
11
12
1
1
22
2
2
1
2
0
( )
,
,
...,
0
0
n
n
T
n
nn
n
u
u
u
x
u
u
x
f
A
x
x
x
u
x
=
=
x
x
x
U is called the
upper triangular coefficient matrix
of the quadratic form.
For example,
[
]
1
1
2
3
1
2
3
2
3
1
3
2
(
,
,
)
,
,
1
3
4
4
2
1
x
f x
x
x
x
x
x
x
x
=
−
−
It may be written as
[
]
2
2
2
1
2
3
1
1
2
1
3
2
2
3
3
1
T
1
2
3
2
3
(
,
,
)
2
6
3
2
1
2
6
,
,
0
3
2
U
0
0
1
f x
x
x
x
x x
x x
x
x x
x
x
x
x
x
x
x
=
+
+
+
+
+
=
=
x
x
Since
x
i
x
j
= x
j
x
i
, it may be rewritten as