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 n-ary 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
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