pletely characterize those properties of quadratic forms in
R
n
which do not depend
on the choice of coordinates. The matrix formulation of the Inertia Theorem reads:
Any symmetric matrix
Q
can be transformed by
Q
mapsto→
C
t
QC
with invertible
C
to one and exactly one of the diagonal forms
I
p
0
0
0
−
I
q
0
0
0
0
.
Exercises
3
.
5
.
1
.
(a) For each of the following quadratic forms
Q
(
x
), write down the corresponding symmetric
matrix
Q
= [
q
ij
], the symmetric bilinear form
Q
(
u
,
v
); then, following the proof of the theorem,
transform the quadratic form to
P
±
x
2
i
and find the inertia indices:
Q
=
x
2
1
+
x
1
x
2
+
x
3
x
4
, Q
=
x
1
x
2
+
x
2
x
3
+
x
3
x
1
, Q
=
x
2
1
+ 2
x
1
x
2
+ 2
x
2
2
+ 4
x
2
x
3
+ 5
x
2
3
, Q
=
x
2
1
−
4
x
1
x
2
+ 2
x
1
x
3
+ 4
x
2
2
+
x
2
3
.
(b) Let
Q
(
x
) be a positive quadratic form. Prove that the determinant det[
q
ij
] of the corre-
sponding symmetric matrix is positive.
(c) Given a quadratic form
Q
(
x
), denote ∆
1
=
q
11
,
∆
2
=
q
11
q
22
−
q
12
q
21
, ...,
∆
n
= det[
q
ij
]
the
k
×
k
-minors ∆
k
= det[
q
ij
]
,
1
≤
i,j
≤
k
of the corresponding symmetric matrix. They are
called
principal minors
of the symmetric matrix [
q
ij
]. Prove that if
Q
is positive then ∆
1
,...,
∆
n
>
0.
(d) Suppose that all principal minors of a quadratic form are non-zero. Following the proof
of the theorem, show that the basis
f
1
,...,
f
n
diagonalizing the quadratic form can be chosen in
such a way that
f
1
is proportional to
e
1
,
f
2
is a linear combination of
e
1
and
e
2
,
f
3
is a linear
combination of
e
1
,
e
2
,
e
3
, etc.
(e) Deduce that a symmetric matrix
Q
with non-zero principal minors can be written as
Q
=
U
t
DU
where
U
is an invertible upper-triangular matrix, and
D
is a diagonal matrix with
the diagonal entries
±
1.
(f) Deduce the Sylvester theorem:
the negative inertia index
q
of a quadratic form with non-
zero principal minors equals the number of changes of signs in the sequence
∆
0
= 1
,
∆
1
,...,
∆
n
.
(g) Test the Sylvester theorem in examples of Exercise (a).
(h) Find inertia indices of the quadratic form
P
i
negationslash
=
j
x
i
x
j
.
(i) Classify surfaces in
R
3
given by equations
F
(
x
1
,x
2
,x
3
) = 0, where
F
is a polynomial of
degree
≤
2, up to linear inhomogeneous changes of coordinates
x
i
=
c
i
1
x
′
1
+
c
i
2
x
′
2
+
c
i
3
x
′
3
+
d
i
, i
= 1
,
2
,
3
.
q
q
q
q
q
q
q
11
12
21
22
1n
n1
nn
∆
∆
∆
∆
∆
1
2
3
n-1
n

3.5. QUADRATIC FORMS
103
3.5.2. Least square fitting to data.
In pure mathematics, there is exactly
one straight line through any two distinct points in the plane. In empirical sciences,
one tends to suspect a linear dependence
y
=
α
+
βx
in any cloud of experimental
points graphed on the plane and to explain the deviation of the points from the line
by natural experimental errors.
The problem we will discuss here is how to find
α
and
β
which make the line
y
=
α
+
βx
fit the experimental data with minimal
error.