6Quadratic forms and conics
IA Vectors and Matrices
6.1 Quadrics and conics
6.1.1 Quadrics
Definition
(Quadric)
.
A quadric is an
n
-dimensional surface defined by the
zero of a real quadratic polynomial, i.e.
x
T
Ax + b
T
x + c = 0,
where
A
is a real
n ×n
matrix,
x, b
are
n
-dimensional column vectors and
c
is a
constant scalar.
As noted in example, anti-symmetric matrix has
x
T
Ax
= 0, so for any
A
,
we can split it into symmetric and anti-symmetric parts, and just retain the
symmetric part S = (A + A
T
)/2. So we can have
x
T
Sx + b
T
x + c = 0
with S symmetric.
Since
S
is real and symmetric, we can diagonalize it using
S
=
QDQ
T
with
D diagonal. We write x
0
= Q
T
x and b
0
= Q
T
b. So we have
(x
0
)
T
Dx
0
+ (b
0
)
T
x
0
+ c = 0.
If
S
is invertible, i.e. with no zero eigenvalues, then write
x
00
=
x
0
+
1
2
D
−1
b
0
which shifts the origin to eliminate the linear term (
b
0
)
T
x
0
and finally have
(dropping the prime superfixes)
x
T
Dx = k.
So through two transformations, we have ended up with a simple quadratic form.
6.1.2 Conic sections (n = 2)
From the equation above, we obtain
λ
1
x
2
1
+ λ
2
x
2
2
= k.
We have the following cases:
(i) λ
1
λ
2
>
0: we have ellipses with axes coinciding with eigenvectors of
S
.
(We require
sgn
(
k
) =
sgn
(
λ
1
, λ
2
), or else we would have no solutions at all)
(ii) λ
1
λ
2
< 0: say λ
1
= k/a
2
> 0, λ
2
= −k/b
2
< 0. So we obtain
x
2
1
a
2
−
x
2
2
b
2
= 1,
which is a hyperbola.
(iii) λ
1
λ
2
= 0: Say
λ
2
= 0,
λ
1
6
= 0. Note that in this case, our symmetric
matrix S is not invertible and we cannot shift our origin using as above.
From our initial equation, we have
λ
1
(x
0
1
)
2
+ b
0
1
x
0
1
+ b
0
2
x
0
2
+ c = 0.
We perform the coordinate transform (which is simply completing the
square!)
x
00
1
= x
0
1
+
b
0
1
2λ
1
x
00
2
= x
0
2
+
c
b
0
2
−
(b
0
1
)
2
4λ
1
b
0
2
to remove the x
0
1
and constant term. Dropping the primes, we have
λ
1
x
2
1
+ b
2
x
2
= 0,
which is a parabola.
Note that above we assumed
b
0
2
6
= 0. If
b
0
2
= 0, we have
λ
1
(
x
0
1
)
2
+
b
0
1
x
0
1
+
c
=
0. If we solve this quadratic for
x
0
1
, we obtain 0, 1 or 2 solutions for
x
1
(and x
2
can be any value). So we have 0, 1 or 2 straight lines.
These are known as conic sections. As you will see in IA Dynamics and Relativity,
this are the trajectories of planets under the influence of gravity.