16B. Quadratic forms
1
Quadratic forms
As a useful application of the Spectral Theorem,
consider the problem of identifying conic section
curves in the plane and quadric surfaces in space
from their equations.
Recall that curves with equations
y = ax 2
are parabolas;
x2 y2
2
2
+
=
1
are
ellipses
(circles,
if
a
=
b
);
2
2
a b
x2 y2
−
= 1 are hyperbolas;
2
2
a b
€
and all three types of curve are called conic
sections, because they are slices of a cone:
16B. Quadratic forms
2
(Intersections between the cone u 2 = v 2 + z 2 and
planes of the form au + bv + cw = d are curves on
these planes whose equations have the general
form of a quadratic equation in two variables:
€
€ 2
(*)
Ax + Bxy + Cy 2 + Dx + Ey + F = 0
in an (x,y) coordinate system on those planes.) The
equations on the previous page are called reduced
€
equations
for the respective curves.
We can also represent these curves by considering
instead the homogeneous quadratic form
ϕ (x, y) = Ax2 + Bxy + Cy 2 ,
called homogeneous since each term has the same
total degree (the sum of the degrees in each
€ and quadratic since this total degree is
variable)
always 2. The conic section (*) can then be
interpreted as the graph of the equation
ϕ (x, y) + Dx + Ey + F = 0.
But the quadratic form ϕ (x, y) has a representation
2
in terms
of
the
inner
product
on
R
:
€
€
€
16B. Quadratic forms
3
ϕ (x, y, z) = Ax 2 + Bxy + Cy 2
 A 1 B  x 
2
 
= x y 
1B
C  y 
2
(
)
= X tr MX
= X,T(X)
where T(X) is the endomorphism on R 2 whose
matrix
€ in the standard basis {E1, E2 } is the
symmetric matrix M. In fact, since M is
symmetric, T must be self-adjoint.
Hence, it is
€
diagonalizable: there is an orthonormal basis
€
{F1, F2 } for R 2 in which T has the diagonal matrix
e 0 
D=  1

 0 e2 
€
€
where e1,e2 are the eigenvalues of T. If X has
coordinates €(u,v) relative to {F1, F2 }, then the
quadratic form can be expressed as
€
ϕ (u,v) = X,T(X)
€
e 0 u 
= u v1
 
 0 e2  v 
(
)
= e1u 2 + e 2v2
€
16B. Quadratic forms
4
Also, the change of basis matrix P whose columns
are the vectors F1,F2 satisfies M = PDP −1 . That is,
x
u 
  = P ⋅  .
y
 v €
€
In particular, substituting expressions for x and y
in terms of u and v allows us to put the equation
ϕ (x, y) + Dx +€Ey + F = 0 in the form
e1u 2 + e2v 2 + D′u + E′v + F = 0 .
(**)
€
The axes of the (u,v) coordinate system for R 2 are
called
€ the principal axes for the conic section.
Now since e1 and e2 are not both zero
€ (else the
equation is not quadratic at all), we obtain three
cases:
(1)€Only one of the eigenvalues is nonzero. By
switching the u and v variables, if necessary, we
may assume that e2 = 0. We can then complete the
square in u to write (**) in the form
€
e1(u − h)2 = l(v − k)
for certain values of h, k and l. It follows that a
€
16B. Quadratic forms
5
translation of the (u,v) coordinate system given by
X = u − h, Y = v − k , brings the equation to the still
simpler form
€
e1X 2 = lY .
If l = 0, then the equation simplifies to X 2 = 0, the
degenerate€conic whose graph corresponds to (two
copies of) the Y-axis. Otherwise, l ≠ 0 and our
equation has the reduced form of a parabola.
€
(2) Both eigenvalues have the same sign. Here,
multiplication of (**) by –1, if necessary, allows us
to assume that both eigenvalues are positive.
Complete the square in both variables u and v to
obtain the equation
e1(u − h)2 + e2 (v − k)2 = l
for certain values of h, k and l, and the translation
X = u − h, Y = v − k brings the equation to the
€ form
simpler
€
e1X 2 + e2Y 2 = l .
If l is negative, the equation has no (real) graph;
and if l = 0, the graph is the single point consisting
€ in the (X,Y) coordinate system. But if l
of the origin
16B. Quadratic forms
6
is positive, then we have neither of these
degenerate forms; instead, the equation has the
reduced form of an ellipse (a circle, when e1 = e2 ).
(3) The eigenvalues have opposite sign. Completing
the square in both u and v allows us to write (**) in
€
the form
e1(u − h)2 + e2 (v − k)2 = l
for certain values of h, k and l, and the translation
X = u − h, Y = v − k brings the equation to the
€ form
simpler
€
€
€
e1X 2 + e2Y 2 = l .
If l = 0, then the equation takes the form
2
|e1 |X 2 =|e€
2 |Y ; taking square roots, we get
Y = ± aX , the equation of a degenerate conic
consisting of a pair of lines of opposite slope
intersecting at (h,k) in the (u,v) coordinate system.
If l has the same sign as e1, then, as e2 has the
opposite sign, the equation has the reduced form of
a hyperbola. If l has the opposite sign as e1, then e2
has the same sign
the X and Y axes
€ and switching
€
brings the equation to the same reduced form of a
hyperbola.
€
€
16B. Quadratic forms
7
We summarize the details of this discussion with
the following
Theorem [Principal Axis Theorem for Conics]
The conic section with equation
Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0
is a (possibly degenerate) parabola, ellipse, or
hyperbola depending on the nature of the
€
eigenvalues of the self-adjoint map T on R 2
obtained from the quadratic form
ϕ (x, y) = Ax2 + Bxy +€
Cy 2
 A 1 B x 
2 
= x y  1
 y 
B
C
2

(
)
= X,T(X)
Specifically, if the eigenvalues of T are e1,e2 , the
(possibly
degenerate) conic is a parabola when one
€
eigenvalue equals 0, an ellipse when both have the
same sign (a circle when they are equal), and a
€
hyperbola when they have opposite
sign. In fact,
since the discriminant Δ = B 2 − 4AC of the
quadratic form satisfies
€
16B. Quadratic forms
− 14
8
A
Δ = det  1
2 B
e

B
0
1
 = det 
 = e1e2 ,
C 
 0 e2 
1
2
the (possibly degenerate) conic is a parabola when
Δ = 0, an ellipse when Δ < 0 and a hyperbola when
€ Δ > 0. //
€
€
€
These ideas can also be carried up into 3-space in
order to classify the graphs of the 2-dimensional
quadric surfaces, defined by a quadratic equation
in the three space variables:
Ax 2 + By 2 + Cz 2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0
As in the situation for conics, we can infer the
general structure of the quadric surface by
considering the quadratic form
€
ϕ (x, y, z) = Ax 2 + By 2 + Cz 2 + Dxy + Eyz + Fxz
(
= x
y
A

z  12 D
1
2 F
= X,T(X)
€
)
1D
2
B
1E
2
1F
2
 x 
1E y
 
2

C  z 
16B. Quadratic forms
9
where T is the self-adjoint endomorphism on R 3
whose symmetric matrix is
A

M =  12 D
1
2 F
1D
2
B
1E
2
1F
2
€
1E .

2

C 
By the Spectral Theorem, T is diagonalizable, so
3
there is an orthonormal
basis
{F
,
F
,
F
}
for
R
1
2
3
€
comprised of eigenvectors for the respective
eigenvalues e1,e2 ,e3 of T. Then, if X has coordinates
(u,v,w) relative to {F1,€F2, F3 }, the quadratic
form
€
can be expressed as
€
ϕ (u,v,w) = X,T(X)
€
e1 0 0  u 


= u v w  0 e2 0  v 
 0 0 e w 
3

(
)
= e1u2 + e2v 2 + e3w 2
Also, the change of basis matrix P whose columns
−1
are
the
vectors
F
,F
,F
satisfies
M
=
PDP
. That
1 2
3
€
is,
€
€
16B. Quadratic forms
10
x
u 
 y  = P ⋅  v .
 
 
z
 
w 
Substituting expressions for x, y and z in terms of u,
v and w allows us to put the equation of the quadric
€ form
surface in the
(†) e1u 2 + e2v 2 + e 3w2 + G ′u + H ′v + I ′w + J ′ = 0 .
€
The axes of the (u,v,w) coordinate system for R 3 are
the principal axes for the quadric surface (and point
in the directions of the eigenvectors).
€
An analysis similar to the one we did above for
conic sections can now be done to bring the
equation (†) into one of 17 reduced forms for
quadrics. Many of them are degenerate
quadrics:
16B. Quadratic forms
11
x2
x2
2
+
a2
y2
a b2
x 2 y 2 z2
+
+
2
2
a b
c2
x2
€
2
a
x2
a
€
€
€
€
€
2
x2
a
2
2
b
y2
b
2
+
c
2
=0
=0
= 0 corresponds to a single point (0,0,0);
corresponds to a single line (z-axis);
corresponds to a pair of coincident
=1
corresponds to a pair of parallel planes
(x = a and x = –a); and
x2
a
+
2
z2
planes (the single yz-plane);
x2
a
+
y2

= −1


= −1 are empty graphs in R 3 ;


= −1

€
2
−
y2
2
b
=0
corresponds to a pair of intersecting
planes ( y = ±(b/a)x ).
16B. Quadratic forms
12
There are other (not quite so) degenerate quadrics,
called cylinders:
z = ax 2
is a parabolic cylinder;
€
x2
a
2
+
y2
b
2
=1
is an elliptic cylinder;
=1
is a hyperbolic cylinder.
€
x2
a
€
2
−
y2
2
b
16B. Quadratic forms
13
This leaves the six nondegenerate quadrics:
z=
x2
a
2
+
y2
is an elliptic paraboloid;
2
b
€
z=
x2
a
2
−
y2
b
is a hyperbolic paraboloid;
2
€
x2
a
€
2
+
y2
b
2
−
z2
c
2
=0
is an elliptic cone;
16B. Quadratic forms
x2
a
2
+
y2
b
2
+
14
z2
c
2
=1
is an ellipsoid;
=1
is a hyperboloid of one sheet;
€
x2
a
2
+
y2
b
2
−
z2
c
2
€
x2
a
€
2
+
y2
b
2
−
z2
c
2
= −1 is a hyperboloid of two sheets.