1.2. Representation and Equivalence; Matrix Connections; Discriminants 5
u and v to get a representation n — f. For instance, the representation
5 = l
2
+
22
gives
:;)

(?  i )
Q

(i?
as a solution to 5 =
17x2
+ 9Axy +
130y2.
Also note that since T G GL2(Z),
we can use the inverse of this transforming matrix to express u, v as integral
linear combinations of x,y. This sets up a bijection between the represen
tations
n — f and n — q.
Quadratic forms / and g related in this way via invertible matrices over a
ring R are said to be equivalent over R. As the example suggests, equiva
lent forms represent the same elements, and they represent them the same
number of times.
Now we will describe the above process in greater generality. Given
a quadratic form q = Y^ii=i
with A = (dij) G Mn(F); if each
Xi is replaced by a linear combination of xi,..., xn with coefficients in i?,
then the result is a new quadratic form h. More explicitly, given a matrix
T = (tij) G Mn(R), replacing (in q) each X{ by Y^j=i Uj%i yields a new
quadratic form h = h(xi,..., xn) with associated Gram matrix
fTAT.
We
say h is represented by q over i?, denoted h — q. It is useful to observe
R
that q can be viewed as the result of a matrix product:
] =
lxAx,
with x =
\ Xn J
(
X l
\
\
Xn
)
q(xi,...,xn) = (xi,...,xn)A
Thus
h(x) =
lxlTATx
= \Tx)A{Tx) = q(Tx).
It follows that if a —• h and h — q then also a — q.
R
Ft
R
Definition 1.8. Let q,h be nary quadratic forms over F, with respective
Gram matrices A, C. We say q and h are equivalent over R and are in
the same class over R if there is a matrix T G GLn(R) such that h —• q
R
via T. That is, h(x) = q(Tx) with T an invertible i?matrix. We denote
equivalence of quadratic forms h and q over Rby h = q. (If JR is understood
R
from the context, to avoid visual clutter we will usually denote equivalence
by h = q and representation by h — q.)
Remark 1.9. A matrix T G GLn(R) is called a unimodular i?matrix.
It determines an automorphism of
Rn
via matrix multiplication. So from
the equation h{x) = q(Tx) we see that equivalent forms represent the same