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