4 1. A Brief Classical Introduction
1.2. Representation and Equivalence; Matrix Connections;
Discriminants
The theory of quadratic forms underwent a dramatic change of style and
perspective in the Twentieth Century; but before getting on with that (in
the next chapter), in keeping with the tenor of this brief look at history
we will first consider—without formal proof—an example involving binary
quadratic forms in the style of Lagrange. Lagrange observed that if each
variable in a given quadratic form is replaced by a linear combination of
variables, then the result is again a quadratic form. It may turn out that
representations by the transformed form are easier to determine than those
of the original form; and then one can carry those representations back and
produce representations by the original form. With this general outline, our
goal here is to determine what integers n are represented over Z by the form
/ =
17x2
+ 94xy +
l30y2.
In particular, can we find an explicit representation 5 f?
( 17 47 \
Consider the Gram matrix A I } for / . First observe that
a sequence of elementary row and column operations for Z-matrices (each
time following a row operation with the corresponding column operation
to get a symmetric resulting matrix) reduces A to a matrix in which the
off-diagonal entry is smaller than the diagonal entries:
17 47 \ / 17 47 \ / 17 13 \ / 4 3
47 130 J ^ V 13 36 J V 13 10 y V 13 10
1 3 \ / l 3 \ / 1 0
x3 I O J ^ O i ; ^ v ° i
3 —11 \
Here T = ( is the product of the elementary matrices—listed
- 1 4
/
from left to right in their order of use—corresponding to the column opera-
tions, and we have observed that
lTAT
f J. Now use the matrix T
to dictate a change of variables: in our given form, make the substitutions
x \— 3u 11^ and y i— u + 4v.
Then upon carrying out all the arithmetic we get
f(x,y) = g(u,v) =
u2
+
v2.
(In other words, we have gotten a new quadratic form with Gram matrix
lTAT.)
But we know from Fermat's result cited earlier exactly which inte-
gers are represented by g. And given explicit values for u and v that solve
the equation n =
u2
+
v2,
we can use the equations relating x and y to
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