3.2.1 Quadratic forms: definitions
A quadratic form in many variables is the sum of several terms, each of which is a constant times the product of exactly two variables.
- Definition
-
A quadratic form in n variables is a function Q for which
| Q(x1, ..., xn) |
= |
b11x2
1 + b12x1x2 +
... + bijxixj + ... + bnnx2
n |
|
|
= |
∑n
i=1 ∑n
j=1 bijxixj |
for all (x1, ..., xn), where bij for i = 1, ..., n and j = 1, ..., n are constants.
- Example 3.2.1.1
-
The function
| Q(x1, x2) |
= |
2x2
1 + 4x1x2 − 6x2x1 −
3x2
2 |
is a quadratic form in two variables.
We can write the quadratic form in this example as
| Q(x1, x2) |
= |
(x1, x2)· |
 |
2 |
4 |
 |
| −6 |
−3 |
|
· |
|
x1 |
|
| x2 |
|
. |
Because 4
x1x2 − 6
x2x1 = −2
x1x2, we can alternatively write it as
| Q(x1, x2) |
= |
(x1, x2)· |
|
2 |
−1 |
|
| −1 |
−3 |
|
· |
|
x1 |
|
| x2 |
|
. |
In this way of writing the quadratic form, the matrix is
symmetric.
We can in fact write any quadratic form as
Q(x) = x'Ax
where
x is the column vector of
xi's and
A is a
symmetric n ×
n matrix for which the (
i,
j)th element is
aij = (1/2)(
bij +
bji). The reason is that
xixj =
xjxi for any
i and
j, so that
| bijxixj + bjixjxi |
= |
(bij + bji)xixj |
|
|
= |
(1/2)(bij + bji)xixj + (1/2)(bji + bij)xjxi. |
- Example 3.2.1.2
-
Let Q(x1, x2, x3) = 3x2
1 + 3x1x2 − x2x1 + 3x1x3 + x3x1 + 2x2x3 +
4x3x2 − x2
2 + 2x2
3. Then, for example, b11 = 3, b12 = 3, and b21 = −1, so that a11 = 3 and a12 = a21 =
(1/2)(3 − 1) = 1. We have Q(x) = x'Ax where
| A = |
 |
3 |
1 |
2 |
 |
| 1 |
−1 |
3 |
| 2 |
3 |
2 |
|
. |
Subsequently, when representing a quadratic form as x'Ax we always take the matrix A to be symmetric.
