6Quadratic forms and conics
IA Vectors and Matrices
6 Quadratic forms and conics
We want to study quantities like
x
2
1
+
x
2
2
and 3
x
2
1
+ 2
x
1
x
2
+ 4
x
2
2
. For example,
conic sections generally take this form. The common characteristic of these is
that each term has degree 2. Consequently, we can write it in the form
x
†
Ax
for some matrix A.
Definition
(Sesquilinear, Hermitian and quadratic forms)
.
A sesquilinear form
is a quantity
F
=
x
†
Ax
=
x
∗
i
A
ij
x
j
. If
A
is Hermitian, then
F
is a Hermitian
form. If A is real symmetric, then F is a quadratic form.
Theorem. Hermitian forms are real.
Proof.
(
x
†
Hx
)
∗
= (
x
†
Hx
)
†
=
x
†
H
†
x
=
x
†
Hx
. So (
x
†
Hx
)
∗
=
x
†
Hx
and it is
real.
We know that any Hermitian matrix can be diagonalized with a unitary
transformation. So
F
(
x
) =
x
†
Hx
=
x
†
UDU
†
x
. Write
x
0
=
U
†
x
. So
F
=
(x
0
)
†
Dx
0
, where D = diag(λ
1
, ··· , λ
n
).
We know that x
0
is the vector x relative to the eigenvector basis. So
F (x) =
n
X
i=1
λ
i
x
0
i

2
The eigenvectors are known as the principal axes.
Example.
Take
F
= 2
x
2
−
4
xy
+ 5
y
2
=
x
T
Sx
, where
x
=
x
y
and
S
=
2 −2
−2 5
.
Note that we can always choose the matrix to be symmetric. This is since
for any antisymmetric
A
, we have
x
†
Ax
= 0. So we can just take the symmetric
part.
The eigenvalues are 1
,
6 with corresponding eigenvectors
1
√
5
2
1
,
1
√
5
1
−2
.
Now change basis with
Q =
1
√
5
2 1
1 −2
Then x
0
= Q
T
x =
1
√
5
2x + y
x − 2y
. Then F = (x
0
)
2
+ 6(y
0
)
2
.
So F = c is an ellipse.