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Chapter XI Quadratic Forms
Chapter XI Quadratlc Forms
1. The set of matrices P,,P,' In this chapter we resume our study of quadratic forms, which was begun in Chapter IV. We shall consider the set of real symmetric positive-definite matrices with respect to congruence transformations over 2, or the set of hermitian positive definite matrices with respect to congruence transformations over certain euclidean rings of complex numbers. We first consider the set PD of all real n x n symmetric positive definite matrices, and its subset Pn',of all members of Pnof determinant 1.
2. An eigenvalue inequality Let A E P.. Then the eigenvalues of A are positive. We denote the smallest eigenvalue of A by p = p(A) and the largest by A = A(A), so that 0 < p A. Then if x is any real n x 1 vector, Thus if x # 0,
pxTx XTX
xTAx
lxTx
< (l/p)xTAx 201
202
XI
QUADRATIC FORMS
It follows that the number of integral vectors x such that xTAx I c, where c is any positive constant, does not exceed the number of solutions of the inequality
I clp and so is finite. This implies that A possesses an arithmetic minimum, which we denote by m(A), as was done in Chapter IV. Thus m(A) = min xTAx XTX
XEV. X#O
where V,, is the set of integral n x 1 vectors. Then it follows precisely as in Chapter IV that (1)
m(A) I (+)'n-1'/2
dll",
d
= det(A)
The results of Chapter VI give the bound (2)
m(A) I(4/n)r[l
+ (n/2)]2/"d l / " ,
d = det(A)
which is better than (1) for large n.
3. The Hermite constant 7. The exact nature of d-'l"m(A) is not yet known. Since m(IA) = Im(A), I > 0, we may confine our study of m(A) to A belonging to P,,'. Then we define the Hermite constant y,, by Thus y,, 2 m(A) for any A of P,,', and because of (l), (4)
It may be proved that there is a matrix A of P,,'such that y,, = m(A); see [60],for example. However we do not actually require this result in what follows. 4. An inequality of Mordell The theorem that follows is due to Mordell [35] and provides a useful inequality satisfied by y,,.
Theorem XI.1 (Mordell). The Hermite constant yn satisfies
4. An Inequality of Mordell Y,
(5)
203
1,
=2 / 0 9 n 2 3 1, and y 2 = 2 / n because of (4), and because
=
Y,,5 7t--)I(,,l 2)
(6) Proof. Certainly y1 = of the matrix
YZ
which belongs to P2' and satisfies m(A) = 2 1 0 . Suppose then that n 2 3. Let A be any matrix of P,,'.Then A-' also belongs to P,,'.Set A = (a,,), A-' = (Ai,). By first replacing A-I by U T A - l U [so that A is replaced by U - l A ( U - l ) T ] ,where U is some suitable matrix of GL(n, Z), we may assure that A , = m(A-') Next, by transformations of the type 1 0 ]A" O]]. O ]A41 O O V T o v 0 V-1 0 (V-1)T where V E GL(n - 1, Z), we may retain A , and make aZ2= m(A where A , is the principal minor matrix of A obtained by deleting the first row and column of A. Then A I E P,- and det(A = A l . Thus az2I yn-lAf:"-l) = yn- ,{m(A- ,)}l'(n- 1) Since aZ2is a value assumed by A for a vector of Vn,we must also have
[
It follows that
]
['
M A ) 5 a22
(7) m(A) 5 yn-l{m(A-l)}l/(n-l) Reversing the roles of A and A-' we also have m(A- 1) < y,- l{m(A)},/("-1 ) (8) (7) and (8) together imply that
(9) m(A) I y:--,,)'-) Since (9) holds for any element of Pn',we may take the supremum of both sides for all A in P,'.The result is that Yn
(n- I )/(nIYn-1
2)
which is what we wished to prove. This concludes the proof.
204
XI
QUADRATIC FORMS
5. Further inequalities for y. An inequality in the other direction was given in [42], and is as follows : Theorem XI.2. The Hermite constant yn satisfies
Y:T: 2 YmmYn"
(1 0)
and thus Yn 2 y!r1"'"
(1 1)
Proof. We use the fact that if A
Pn,B
E
E
P,, then
m(A i S ) = min(m(A), m(B))
Choose E > 0 arbitrarily. Then there are matrices A , B such that A E P,,B E P,, and det(A) = 1, det(B) = (Y,,/Y,)~, Thus m(B) 2 y n - E also. Hence But also
m(A
4-B ) = min(m(d),
m(A i B) 5 y,+,(det(A It follows that (12)
m(A) 2 Yn - 6, m(B) 2 y,(det(B))l/m - E
m(B)) 2 y, - E
-i-B))ll(m+n) =-Y r n + n ( Y n / Y n ) m / ( r n + n )
Yrn +n(Yn/Ym)m'(m
+n)
>
Yn
--E
Since (12) holds for all positive 6, we have Ym+n(Yn/Ym)n/(m
+ n,
>
Yn
which is just (10); (11) follows from (10) by choosing replacing n by IZ - 1. This completes the proof.
M =1
and
Notice that (13) Y. 2 1 since m(Z) = 1 and Z E Pn'. Formula (2) shows that yn < Cn,for some suitable positive constant C'Minkowski showed by the geometry of numbers that also y. > cn, for some other constant c. Thus
logyn/logn --+ 1 as n -+
00
7 . Factorization of Hermitian Positive Definite Matrices
205
Whether yn/n approaches a limit as n approaches 00 is unknown. The actual values of y. are known for 1 < n < 8, and are all of the form 2a3b,where a, b are rationals. 6. A factorization of an arbitrary complex matrix
We are now going to develop bounds for the number of congruence classes over 2 of the n x n symmetric positive definite matrices over 2. For this purpose we require some elementary information about the matrices of P.. We first prove Lemma XI.1. Let A may be written as
= (ai,)
be any n x n complex matrix. Then A A
=
UT
where U is unitary and T is upper triangular. If A is real, U may be chosen orthogonal and T real.
-
Proof. Let C, = [a,laz, . . anJTbe the first column of A . If C, = 0, choose 17, = I. If C , # 0, choose U,as a unitary matrix whose first column is (1 /c)C,, where
Then in either case, the entries of the first column of U,*A are all 0, except possibly for the (1, 1) element. This construction may now be repeated using operations of the form I , i u,, I , i u,, * * * I n - 2 i u"-l where U,,U,, . . . , UE-l are unitary matrices, until A is reduced to upper triangular form. Furthermore it is clear that if A is real, each U, may be chosen real, so that U may be chosen orthogonal, whence T must also be real. This completes the proof. 9
7. Factorization of hermitian positive definite matrices Now let A be any element of Pn.By the results of Chapter V, we may write A = UTDU where U is orthogonal and D
=
diag(d,, d,, . . . , d,,), where d, > 0.
206
XI QUADRATIC FORMS
Let e, be the positive square root of d, and put E = diag(e,, e,, . . . ,en), B = UTEU.Then B E P,,and A = B2. Thus we have shown that every element of P. has a square root in P.. Now by Lemma XI.],B may be written as B = VT, where V is orthogonal and T is upper triangular. Then A = B2 = BTB = T'T
Thus we have also shown that A may be written as TTT,where T is real
and upper triangular. We may clearly choose the diagonal elements of T positive. We will need to know these diagonal elements in terms of the elements of A . For each r such that 1 r 5 n - 1, write Tin partitioned form as
<
where U,is an r x r upper triangular matrix, V , is r x (n - r), and W, is an (n - r) x (n - r) upper triangular matrix. Then A=TTT=[
ur'ur ~
:]
Let A, be the principal minor matrix of A obtained by striking out the last n - r rows and columns of A. Then A,
It follows that if
=
U,'U,
d, = det(A,), 1
t,,
n
-
1
l l r l n
d, = t l 2 t Z 2... f r 2 ,
1
5 r
-1
Thus if we define then
do = I ,
tr2 = drIdr-,,
dn = det(A)
l l r l n We can modify this construction so that T is lower triangular instead of upper triangular, with corresponding results about the
207
8. Bounds for Cofactors
diagonal elements of T. We collect this information into a lemma: Lemma XI.2. Let A be any element of Pa. Then A may be written as B2,where B also belongs to P., or as TTT,where T where T is lower triangular with positive diagonal elements t,, t , , . . . ,t,, whose values are given by
t r 2 = dr-,/dr, 1
(14)
8. Bounds for cofactors
The next lemma is actually implicit in the proof of Theorem XI.1. Lemma XI.3. Let A belong to P.. Then after a suitable congruence transformation, d, satisfies
d , 5 y,, dtn-I)/" Proof. We apply a congruence transformation to A-' so that the (1, 1) element of the resulting matrix is its arithmetic minimum, and make the corresponding congruence transformation on A. We continue to denote the resulting matrix by A. Since the (I, 1 ) element of A-I is d , / d , and the determinant of A-I is l / d , we have (15)
d , / d = m(A-') 5 Y,,(l/d)"", Q
< Y. d l - l / n
1 -
Lemma XI.4. A congruence transformation of the form Zr U, E GL(n - r, Z ) , does not change d,, 0 5 s 5 r. Proof.
If 0 5 s I r we have
where V,
r, iu, = z, 4- rr-s4- u, = z, iv,
E
GL(n - s, Z ) . Then the congruence transformation (1, 4-uATA (1, i u,)
replaces A, by VsTA,V,,and so does not change d,
=
det(A,).
i U,,
208
XI
QUADRATIC FORMS
We now obtain the following theorem from these lemmas, which rI n: provides bounds for the numbers d,, 0 I Theorem XI.3. After a suitable congruence transformation the numbers d, satisfy (16)
dnWr I y r + I d;!;?:),
l
Proof. The validity of (16) follows by induction from Lemmas XI.3 and XT.4. Inequality (17) follows from (16) by producting the inequality I/r < l / r dn-r -Yr+l
dl/(r+l) n-r-I
9. Bounds for class numbers
We now use Theorem XI.3 and Lemma XI.2 to derive bounds for the diagonal elements of A , after the congruence transformations described above have been applied. We have A
=
TTT
where T is lower triangular and has positive diagonal elements t , , t,, . . . , t , satisfying (14). As in the reduction of a matrix to Hermite normal form, we can find a lower triangular matrix U E GL(n, 2) such that the congruence transformation UTAU = (TU)T(TU)
leaves the diagonal elements t , unaltered and makes Iti,I
1< j < i I n
As before, we continue to denote the resulting matrices by A and T. Since
air= t,,
+ 2
k=i+l
tii,
1I i4n
we have the bounds
Furthermore, since A is positive definite, we must have (19)
a:,
< %a,,,
1 4 1, j 4 n.
10. Minko wski's Proof of the Finiteness of the Class Number
209
Thus (17), (18), and (19) together imply the following: Theorem XI.4. Let A be any n x n symmetric positive definite matrix over Z of determinant d. Then A is congruent over Z to a matrix B = (blf)such that
i J l l < d l - l + a zk =di k ,
l
where the numbers dk satisfy the inequalities (17). Then because the total number of possible matrices B is finite and does not exceed W(2M - l)(n'-n)/Z < 2 ( n * - n ) / Z M ( n ' + n ) / t where Ills"
This bound is not at all sharp, but it has the advantage of being both general and explicit. Thus we have given an effective criterion for deciding whether or not two n x n symmetric positive definite matrices over Z of the same determinant are congruent over Z. 10. Minkowski's proof of the finiteness of the class number We now consider Minkowski's approach to the finiteness of the class number, which involves a significant generalization of the arithmetic minimum. Let Vn stand for the lattice of n x 1 vectors over Z , and let A be any element of Pn, d = det(A). Then if v,, vz, . , . , v, are any real n x 1 vectors, the inequaiity
is valid, where p = p ( A ) is the least eigenvalue of A . Thus if w
= W(V,,
vz,. . . ,v,)
=
,n r
,=I
(V,TAV,)
and c is any positive number, then the number of solutions of w l c
in vectors v l ,v,, . . ., v, belonging to Vndoes not exceed the number of
210
XI
QUADRATIC FORMS
solutions of the inequality
and hence is finite. It follows that if 1 5 r n, then the problem of finding the minimum of w for a set of r linearly independent vectors belonging to V , is a finite one, and so has a solution. Choose r = n, and suppose that the minimum occurs for the linearly independent vectors v l ,v 2 , ,. ., v, of V,. Let mi = v,TAv,, I I iI n and assume that the vectors are ordered so that
m 1 2 m , 2 . . - >m,
Let V be the matrix
-
[v1, 8 2 , * . %I Then V is a nonsingular integral matrix, and we may write V = UH where U E GL(n,Z) and H is a lower triangular matrix in Hermite normal form, which we write as the matrix of its column vectors =
H
Then and
=
3
[ h l ,h,, . . . ,h,l
v, = Uh,,
l l i l n
mi = h,'UTAUhi,
1
We may replace A by
Ii I n
B = UTAU
with no change in the value of m, or loss of generality. Then in the new formulation we have mi = h,TBh,, 1 Ii n where hlT = [h11 h2, h,, hnjl, * hzT = [ 0 h,, h,, hn21, h,T = [ 0 0 h,, - * * h n J ,
hnT= [ 0
0
0
* *
-
h,,]
10. Minkowski's Proof of the Finiteness of the Class Number
21 1
Let V,,, be the subset of Vn consisting of all vectors x = [ x , , x 2 , Then in the representation above we see that h, has the following property: Of all vectors of V,,,[,h, is one for which h,TBh, is least. We may write B as CTC,where C is a real lower triangular matrix with positive diagonal elements. Put
. . . ,x , , ) of ~ V,,such that x , is the first nonzero coordinate.
y
(20) Then if x
E
=
cx
Vn,[,the first nonzero coordinate ofy = [y,, y 2 , . . . ,y,]'
is
Yi.
Now suppose that x is any nonzero element of V,. Then x E V,,( for some i such that 1 2 i 5 n. Then according to the property of h, mentioned above, xTBx 2 m i Furthermore X ~ B= X y'y = y,' ~l"+~ so that y,, Y?+l '. ynz Now if x is any vector of V,,, put
+
+ + +
+
a
Thus if x
E
VnSi, then
That is, q(x) 2 1 for every vector x of V, other than x = 0. It follows that the ellipsoid (21) 4(x) I 1 has no lattice points in its interior, other than x = 0; and hence by Minkowski's fundamental theorem of Chapter VI, its volume must be 5 2". Now the volume of (21) in the y-space is
and since the Jacobian of the transformation (20) is det(C) = d 1 l 2 , its volume in the x-space is
212
XI QUADRATIC FORMS
Thus we obtain the inequality The diagonal elements of HTBH are m , , m , , . . . , m , . Then Hadamard's determinantal inequality implies that
Since H is in Hermite normal form, the number of possible choices for H i s bounded. Furthermore, if B is an integral matrix, the number of possible choices for HTBH is bounded, since the diagonal elements of HTBH are bounded. It follows that the number of choices for B is bounded. We have therefore proved that every n x n integral symmetric positive-definite matrix A of fixed determinant d is congruent over 2 to one of finitely many matrices B, thus providing another proof of the finiteness of the class number. The case n = 3 is particularly simple. For then
so that det(H) = I . Since H is in Hermite normal form, H must be I. Thus the product of the diagonal elements of B does not exceed
m,m2m3 (36/n2)d< 4d It is known that the constant 36/w2 may be replaced by 2. 11. Some subrings of complex numbers We now turn to the study of certain important subrings of the complex numbers. Let D be a fixed square-free integer > I . Define @2= -B and put D = 1 , 2 mod 4 D = 3 mod 4 0)/2,
+
Then the integers of Q(@ are just the elements of Z(w). These rings are not always principal ideal rings, and so do not properly belong to the
213
12. The Norm Constant
class of rings that we have been considering. However some of them are euclidean rings, and it is these that we will ultimately consider. 12. The norm constant We first investigate the following question: If z is a complex number, determine the value of k, defined by k = m a x min 1 z -
(23)
a€Z[cul
I
&I2
The definition of k is justified, since the elements of Z[w]are the vertices of a point lattice in the complex plane, and any complex number z lies in one of the fundametal parallelograms defined by this lattice. Thus we may assume that z is restricted to belong to the parallelogram P with vertices 0, I , 1 w, w . Then we have
+
k
= max ZEP
min{l z 12, I z
-
1 12, I z - w - 1 12, I z - w 12)
Because of the symmetry, this is the same as k
= max min(1 z ZET
12, I z - 1 12, 1 z - w 11’
where T is the triangle with vertices 0, 1, w. It is not difficult to prove that the maximum is assumed when z is equidistant from 0, 1, w . Hence k is the square of the radius of the circle circumscribing T,and we obtain Lemma XI.5. The number k defined by (23) satisfies (D 1Y4, D- 1,2mod4 k={ (24) ( D 1)2/16D, D G 3 mod4 Thus for the first few values of D we have the table
+
+
D 1 2 3 5 6 7 . 1 0 11
(25)
13
t a + $ a ? : y ? T %
15
#
+ b w be any element of Z [ w ] . Define a2 + Db2, D E 1,2mod4 N(a) = a& = D = 3 mod 4 a2 + ab + ((D+ 1)/4)bZ,
Now let a
=a
Suppose that a, /j are arbitrary elements of Z [ w ] ,
f
0. According
214
XI QUADRATIC FORMS
to the definition of k, we may find an element y of Z[w]such that
Na/S - y ) I k;
that is, such that
N(a - rB) I kWB) A glance at the table (25) and a few brief considerations now imply
Theorem XI.5. The ring Z[w]is a proper euclidean ring with associated function v(a) = N(a) = a&, a E Z [ o ] ,if and only if D = 1,2,3,7,ll We call k the norm constant of Z [ o ] . 13. Hermitian positive definite matrices and the generalized Hermite constant 7 J D ) We are now prepared to study the set Sn of n x n hermitian positive definite matrices A with respect to transformations of the type U*AU
where U E GL(n,Z[w]).Just as in the previous discussion, we can prove that any such matrix A assumes its arithmetic minimum m(A), where now the quadratic form associated with A is x*Ax, x any n x 1 vector. We can define the "generalized Hermite constant" y,(D) by
y
m
= SUP, AEO.
m(4
where S,' is the subset of S,, consisting of all matrices of determinant 1. The preceding proofs go over practically unchanged and we have Theorem XI.6. Let D = 1,2, 3,7, 11. Then the generalized Hermite constant y,(D) satisfies (26)
y l ( D ) = 1,
yz(D) = (1 - k)-'/',
(27)
y,(D) I y"-l(D)("-"""-z) , n 2 3
Thus in particular (28)
m(D)
(I/( 1 - k))'"-')/'
For the cases of interest, we have the following table:
215
Exercises and Problems
(29)
D k
1
2
3
7
11
3
3
3
1:
A
0 2 3 6 +h/TI
(1 - k)-"2
From an examination of this table, we can conclude
Corollary XI.1. Let A be any 2 x 2 hermitian positive-definite matrix of determinant 1 over Z[o]. Then if D = 1, 3, or 7, A must be of the form B*B, where B is a matrix over Z[o]. 14. Some consequences
Some interesting number-theoretic consequences of this corollary may be derived. We take D = 3 as an example. Let p be any prime # 3. Then it is not difficult to prove that the congruence x2
+ xy + y 2 + 1
E
0 modp
always has solutions. Let (x, y ) be a solution, and set x2 1 = mp. Put ~ = [ x + c y
"'"'1
+ xy + y 2 +
m
Then A is a hermitian positive definite matrix of determinant 1 over
Z[o], and consequently must be of the form B*B, where B is a matrix over Z[w].It follows that integers a, b, c, d exist such that p
= a2
+ ab + b2 + c2 + cd + d2
The classical literature on quadratic forms is too extensive to comment on here in any detail. The results of this chapter do indicate a few of the important landmarks, however. The reader is referred to [8] or [20] for a more detailed treatment. EXERCISES AND PROBLEMS
1. Work out the consequencesof Corollary XI. 1 for the two remaining cases. 2. Let N be an arbitrary positive number. Let SN be the set of real 2 x 2 matrices whose elements do not exceed N in absolute value. Show that P2' always contains matrices which are not congruent over Z to any element of SN.
1. The set of matrices P,,P,' In this chapter we resume our study of quadratic forms, which was begun in Chapter IV. We shall consider the set of real symmetric positive-definite matrices with respect to congruence transformations over 2, or the set of hermitian positive definite matrices with respect to congruence transformations over certain euclidean rings of complex numbers. We first consider the set PD of all real n x n symmetric positive definite matrices, and its subset Pn',of all members of Pnof determinant 1.
2. An eigenvalue inequality Let A E P.. Then the eigenvalues of A are positive. We denote the smallest eigenvalue of A by p = p(A) and the largest by A = A(A), so that 0 < p A. Then if x is any real n x 1 vector, Thus if x # 0,
pxTx XTX
xTAx
lxTx
< (l/p)xTAx 201
202
XI
QUADRATIC FORMS
It follows that the number of integral vectors x such that xTAx I c, where c is any positive constant, does not exceed the number of solutions of the inequality
I clp and so is finite. This implies that A possesses an arithmetic minimum, which we denote by m(A), as was done in Chapter IV. Thus m(A) = min xTAx XTX
XEV. X#O
where V,, is the set of integral n x 1 vectors. Then it follows precisely as in Chapter IV that (1)
m(A) I (+)'n-1'/2
dll",
d
= det(A)
The results of Chapter VI give the bound (2)
m(A) I(4/n)r[l
+ (n/2)]2/"d l / " ,
d = det(A)
which is better than (1) for large n.
3. The Hermite constant 7. The exact nature of d-'l"m(A) is not yet known. Since m(IA) = Im(A), I > 0, we may confine our study of m(A) to A belonging to P,,'. Then we define the Hermite constant y,, by Thus y,, 2 m(A) for any A of P,,', and because of (l), (4)
It may be proved that there is a matrix A of P,,'such that y,, = m(A); see [60],for example. However we do not actually require this result in what follows. 4. An inequality of Mordell The theorem that follows is due to Mordell [35] and provides a useful inequality satisfied by y,,.
Theorem XI.1 (Mordell). The Hermite constant yn satisfies
4. An Inequality of Mordell Y,
(5)
203
1,
=2 / 0 9 n 2 3 1, and y 2 = 2 / n because of (4), and because
=
Y,,5 7t--)I(,,l 2)
(6) Proof. Certainly y1 = of the matrix
YZ
which belongs to P2' and satisfies m(A) = 2 1 0 . Suppose then that n 2 3. Let A be any matrix of P,,'.Then A-' also belongs to P,,'.Set A = (a,,), A-' = (Ai,). By first replacing A-I by U T A - l U [so that A is replaced by U - l A ( U - l ) T ] ,where U is some suitable matrix of GL(n, Z), we may assure that A , = m(A-') Next, by transformations of the type 1 0 ]A" O]]. O ]A41 O O V T o v 0 V-1 0 (V-1)T where V E GL(n - 1, Z), we may retain A , and make aZ2= m(A where A , is the principal minor matrix of A obtained by deleting the first row and column of A. Then A I E P,- and det(A = A l . Thus az2I yn-lAf:"-l) = yn- ,{m(A- ,)}l'(n- 1) Since aZ2is a value assumed by A for a vector of Vn,we must also have
[
It follows that
]
['
M A ) 5 a22
(7) m(A) 5 yn-l{m(A-l)}l/(n-l) Reversing the roles of A and A-' we also have m(A- 1) < y,- l{m(A)},/("-1 ) (8) (7) and (8) together imply that
(9) m(A) I y:--,,)'-) Since (9) holds for any element of Pn',we may take the supremum of both sides for all A in P,'.The result is that Yn
(n- I )/(nIYn-1
2)
which is what we wished to prove. This concludes the proof.
204
XI
QUADRATIC FORMS
5. Further inequalities for y. An inequality in the other direction was given in [42], and is as follows : Theorem XI.2. The Hermite constant yn satisfies
Y:T: 2 YmmYn"
(1 0)
and thus Yn 2 y!r1"'"
(1 1)
Proof. We use the fact that if A
Pn,B
E
E
P,, then
m(A i S ) = min(m(A), m(B))
Choose E > 0 arbitrarily. Then there are matrices A , B such that A E P,,B E P,, and det(A) = 1, det(B) = (Y,,/Y,)~, Thus m(B) 2 y n - E also. Hence But also
m(A
4-B ) = min(m(d),
m(A i B) 5 y,+,(det(A It follows that (12)
m(A) 2 Yn - 6, m(B) 2 y,(det(B))l/m - E
m(B)) 2 y, - E
-i-B))ll(m+n) =-Y r n + n ( Y n / Y n ) m / ( r n + n )
Yrn +n(Yn/Ym)m'(m
+n)
>
Yn
--E
Since (12) holds for all positive 6, we have Ym+n(Yn/Ym)n/(m
+ n,
>
Yn
which is just (10); (11) follows from (10) by choosing replacing n by IZ - 1. This completes the proof.
M =1
and
Notice that (13) Y. 2 1 since m(Z) = 1 and Z E Pn'. Formula (2) shows that yn < Cn,for some suitable positive constant C'Minkowski showed by the geometry of numbers that also y. > cn, for some other constant c. Thus
logyn/logn --+ 1 as n -+
00
7 . Factorization of Hermitian Positive Definite Matrices
205
Whether yn/n approaches a limit as n approaches 00 is unknown. The actual values of y. are known for 1 < n < 8, and are all of the form 2a3b,where a, b are rationals. 6. A factorization of an arbitrary complex matrix
We are now going to develop bounds for the number of congruence classes over 2 of the n x n symmetric positive definite matrices over 2. For this purpose we require some elementary information about the matrices of P.. We first prove Lemma XI.1. Let A may be written as
= (ai,)
be any n x n complex matrix. Then A A
=
UT
where U is unitary and T is upper triangular. If A is real, U may be chosen orthogonal and T real.
-
Proof. Let C, = [a,laz, . . anJTbe the first column of A . If C, = 0, choose 17, = I. If C , # 0, choose U,as a unitary matrix whose first column is (1 /c)C,, where
Then in either case, the entries of the first column of U,*A are all 0, except possibly for the (1, 1) element. This construction may now be repeated using operations of the form I , i u,, I , i u,, * * * I n - 2 i u"-l where U,,U,, . . . , UE-l are unitary matrices, until A is reduced to upper triangular form. Furthermore it is clear that if A is real, each U, may be chosen real, so that U may be chosen orthogonal, whence T must also be real. This completes the proof. 9
7. Factorization of hermitian positive definite matrices Now let A be any element of Pn.By the results of Chapter V, we may write A = UTDU where U is orthogonal and D
=
diag(d,, d,, . . . , d,,), where d, > 0.
206
XI QUADRATIC FORMS
Let e, be the positive square root of d, and put E = diag(e,, e,, . . . ,en), B = UTEU.Then B E P,,and A = B2. Thus we have shown that every element of P. has a square root in P.. Now by Lemma XI.],B may be written as B = VT, where V is orthogonal and T is upper triangular. Then A = B2 = BTB = T'T
Thus we have also shown that A may be written as TTT,where T is real
and upper triangular. We may clearly choose the diagonal elements of T positive. We will need to know these diagonal elements in terms of the elements of A . For each r such that 1 r 5 n - 1, write Tin partitioned form as
<
where U,is an r x r upper triangular matrix, V , is r x (n - r), and W, is an (n - r) x (n - r) upper triangular matrix. Then A=TTT=[
ur'ur ~
:]
Let A, be the principal minor matrix of A obtained by striking out the last n - r rows and columns of A. Then A,
It follows that if
=
U,'U,
d, = det(A,), 1
t,,
n
-
1
l l r l n
d, = t l 2 t Z 2... f r 2 ,
1
5 r
-1
Thus if we define then
do = I ,
tr2 = drIdr-,,
dn = det(A)
l l r l n We can modify this construction so that T is lower triangular instead of upper triangular, with corresponding results about the
207
8. Bounds for Cofactors
diagonal elements of T. We collect this information into a lemma: Lemma XI.2. Let A be any element of Pa. Then A may be written as B2,where B also belongs to P., or as TTT,where T where T is lower triangular with positive diagonal elements t,, t , , . . . ,t,, whose values are given by
t r 2 = dr-,/dr, 1
(14)
8. Bounds for cofactors
The next lemma is actually implicit in the proof of Theorem XI.1. Lemma XI.3. Let A belong to P.. Then after a suitable congruence transformation, d, satisfies
d , 5 y,, dtn-I)/" Proof. We apply a congruence transformation to A-' so that the (1, 1) element of the resulting matrix is its arithmetic minimum, and make the corresponding congruence transformation on A. We continue to denote the resulting matrix by A. Since the (I, 1 ) element of A-I is d , / d , and the determinant of A-I is l / d , we have (15)
d , / d = m(A-') 5 Y,,(l/d)"", Q
< Y. d l - l / n
1 -
Lemma XI.4. A congruence transformation of the form Zr U, E GL(n - r, Z ) , does not change d,, 0 5 s 5 r. Proof.
If 0 5 s I r we have
where V,
r, iu, = z, 4- rr-s4- u, = z, iv,
E
GL(n - s, Z ) . Then the congruence transformation (1, 4-uATA (1, i u,)
replaces A, by VsTA,V,,and so does not change d,
=
det(A,).
i U,,
208
XI
QUADRATIC FORMS
We now obtain the following theorem from these lemmas, which rI n: provides bounds for the numbers d,, 0 I Theorem XI.3. After a suitable congruence transformation the numbers d, satisfy (16)
dnWr I y r + I d;!;?:),
l
Proof. The validity of (16) follows by induction from Lemmas XI.3 and XT.4. Inequality (17) follows from (16) by producting the inequality I/r < l / r dn-r -Yr+l
dl/(r+l) n-r-I
9. Bounds for class numbers
We now use Theorem XI.3 and Lemma XI.2 to derive bounds for the diagonal elements of A , after the congruence transformations described above have been applied. We have A
=
TTT
where T is lower triangular and has positive diagonal elements t , , t,, . . . , t , satisfying (14). As in the reduction of a matrix to Hermite normal form, we can find a lower triangular matrix U E GL(n, 2) such that the congruence transformation UTAU = (TU)T(TU)
leaves the diagonal elements t , unaltered and makes Iti,I
1< j < i I n
As before, we continue to denote the resulting matrices by A and T. Since
air= t,,
+ 2
k=i+l
tii,
1I i4n
we have the bounds
Furthermore, since A is positive definite, we must have (19)
a:,
< %a,,,
1 4 1, j 4 n.
10. Minko wski's Proof of the Finiteness of the Class Number
209
Thus (17), (18), and (19) together imply the following: Theorem XI.4. Let A be any n x n symmetric positive definite matrix over Z of determinant d. Then A is congruent over Z to a matrix B = (blf)such that
i J l l < d l - l + a zk =di k ,
l
where the numbers dk satisfy the inequalities (17). Then because the total number of possible matrices B is finite and does not exceed W(2M - l)(n'-n)/Z < 2 ( n * - n ) / Z M ( n ' + n ) / t where Ills"
This bound is not at all sharp, but it has the advantage of being both general and explicit. Thus we have given an effective criterion for deciding whether or not two n x n symmetric positive definite matrices over Z of the same determinant are congruent over Z. 10. Minkowski's proof of the finiteness of the class number We now consider Minkowski's approach to the finiteness of the class number, which involves a significant generalization of the arithmetic minimum. Let Vn stand for the lattice of n x 1 vectors over Z , and let A be any element of Pn, d = det(A). Then if v,, vz, . , . , v, are any real n x 1 vectors, the inequaiity
is valid, where p = p ( A ) is the least eigenvalue of A . Thus if w
= W(V,,
vz,. . . ,v,)
=
,n r
,=I
(V,TAV,)
and c is any positive number, then the number of solutions of w l c
in vectors v l ,v,, . . ., v, belonging to Vndoes not exceed the number of
210
XI
QUADRATIC FORMS
solutions of the inequality
and hence is finite. It follows that if 1 5 r n, then the problem of finding the minimum of w for a set of r linearly independent vectors belonging to V , is a finite one, and so has a solution. Choose r = n, and suppose that the minimum occurs for the linearly independent vectors v l ,v 2 , ,. ., v, of V,. Let mi = v,TAv,, I I iI n and assume that the vectors are ordered so that
m 1 2 m , 2 . . - >m,
Let V be the matrix
-
[v1, 8 2 , * . %I Then V is a nonsingular integral matrix, and we may write V = UH where U E GL(n,Z) and H is a lower triangular matrix in Hermite normal form, which we write as the matrix of its column vectors =
H
Then and
=
3
[ h l ,h,, . . . ,h,l
v, = Uh,,
l l i l n
mi = h,'UTAUhi,
1
We may replace A by
Ii I n
B = UTAU
with no change in the value of m, or loss of generality. Then in the new formulation we have mi = h,TBh,, 1 Ii n where hlT = [h11 h2, h,, hnjl, * hzT = [ 0 h,, h,, hn21, h,T = [ 0 0 h,, - * * h n J ,
hnT= [ 0
0
0
* *
-
h,,]
10. Minkowski's Proof of the Finiteness of the Class Number
21 1
Let V,,, be the subset of Vn consisting of all vectors x = [ x , , x 2 , Then in the representation above we see that h, has the following property: Of all vectors of V,,,[,h, is one for which h,TBh, is least. We may write B as CTC,where C is a real lower triangular matrix with positive diagonal elements. Put
. . . ,x , , ) of ~ V,,such that x , is the first nonzero coordinate.
y
(20) Then if x
E
=
cx
Vn,[,the first nonzero coordinate ofy = [y,, y 2 , . . . ,y,]'
is
Yi.
Now suppose that x is any nonzero element of V,. Then x E V,,( for some i such that 1 2 i 5 n. Then according to the property of h, mentioned above, xTBx 2 m i Furthermore X ~ B= X y'y = y,' ~l"+~ so that y,, Y?+l '. ynz Now if x is any vector of V,,, put
+
+ + +
+
a
Thus if x
E
VnSi, then
That is, q(x) 2 1 for every vector x of V, other than x = 0. It follows that the ellipsoid (21) 4(x) I 1 has no lattice points in its interior, other than x = 0; and hence by Minkowski's fundamental theorem of Chapter VI, its volume must be 5 2". Now the volume of (21) in the y-space is
and since the Jacobian of the transformation (20) is det(C) = d 1 l 2 , its volume in the x-space is
212
XI QUADRATIC FORMS
Thus we obtain the inequality The diagonal elements of HTBH are m , , m , , . . . , m , . Then Hadamard's determinantal inequality implies that
Since H is in Hermite normal form, the number of possible choices for H i s bounded. Furthermore, if B is an integral matrix, the number of possible choices for HTBH is bounded, since the diagonal elements of HTBH are bounded. It follows that the number of choices for B is bounded. We have therefore proved that every n x n integral symmetric positive-definite matrix A of fixed determinant d is congruent over 2 to one of finitely many matrices B, thus providing another proof of the finiteness of the class number. The case n = 3 is particularly simple. For then
so that det(H) = I . Since H is in Hermite normal form, H must be I. Thus the product of the diagonal elements of B does not exceed
m,m2m3 (36/n2)d< 4d It is known that the constant 36/w2 may be replaced by 2. 11. Some subrings of complex numbers We now turn to the study of certain important subrings of the complex numbers. Let D be a fixed square-free integer > I . Define @2= -B and put D = 1 , 2 mod 4 D = 3 mod 4 0)/2,
+
Then the integers of Q(@ are just the elements of Z(w). These rings are not always principal ideal rings, and so do not properly belong to the
213
12. The Norm Constant
class of rings that we have been considering. However some of them are euclidean rings, and it is these that we will ultimately consider. 12. The norm constant We first investigate the following question: If z is a complex number, determine the value of k, defined by k = m a x min 1 z -
(23)
a€Z[cul
I
&I2
The definition of k is justified, since the elements of Z[w]are the vertices of a point lattice in the complex plane, and any complex number z lies in one of the fundametal parallelograms defined by this lattice. Thus we may assume that z is restricted to belong to the parallelogram P with vertices 0, I , 1 w, w . Then we have
+
k
= max ZEP
min{l z 12, I z
-
1 12, I z - w - 1 12, I z - w 12)
Because of the symmetry, this is the same as k
= max min(1 z ZET
12, I z - 1 12, 1 z - w 11’
where T is the triangle with vertices 0, 1, w. It is not difficult to prove that the maximum is assumed when z is equidistant from 0, 1, w . Hence k is the square of the radius of the circle circumscribing T,and we obtain Lemma XI.5. The number k defined by (23) satisfies (D 1Y4, D- 1,2mod4 k={ (24) ( D 1)2/16D, D G 3 mod4 Thus for the first few values of D we have the table
+
+
D 1 2 3 5 6 7 . 1 0 11
(25)
13
t a + $ a ? : y ? T %
15
#
+ b w be any element of Z [ w ] . Define a2 + Db2, D E 1,2mod4 N(a) = a& = D = 3 mod 4 a2 + ab + ((D+ 1)/4)bZ,
Now let a
=a
Suppose that a, /j are arbitrary elements of Z [ w ] ,
f
0. According
214
XI QUADRATIC FORMS
to the definition of k, we may find an element y of Z[w]such that
Na/S - y ) I k;
that is, such that
N(a - rB) I kWB) A glance at the table (25) and a few brief considerations now imply
Theorem XI.5. The ring Z[w]is a proper euclidean ring with associated function v(a) = N(a) = a&, a E Z [ o ] ,if and only if D = 1,2,3,7,ll We call k the norm constant of Z [ o ] . 13. Hermitian positive definite matrices and the generalized Hermite constant 7 J D ) We are now prepared to study the set Sn of n x n hermitian positive definite matrices A with respect to transformations of the type U*AU
where U E GL(n,Z[w]).Just as in the previous discussion, we can prove that any such matrix A assumes its arithmetic minimum m(A), where now the quadratic form associated with A is x*Ax, x any n x 1 vector. We can define the "generalized Hermite constant" y,(D) by
y
m
= SUP, AEO.
m(4
where S,' is the subset of S,, consisting of all matrices of determinant 1. The preceding proofs go over practically unchanged and we have Theorem XI.6. Let D = 1,2, 3,7, 11. Then the generalized Hermite constant y,(D) satisfies (26)
y l ( D ) = 1,
yz(D) = (1 - k)-'/',
(27)
y,(D) I y"-l(D)("-"""-z) , n 2 3
Thus in particular (28)
m(D)
(I/( 1 - k))'"-')/'
For the cases of interest, we have the following table:
215
Exercises and Problems
(29)
D k
1
2
3
7
11
3
3
3
1:
A
0 2 3 6 +h/TI
(1 - k)-"2
From an examination of this table, we can conclude
Corollary XI.1. Let A be any 2 x 2 hermitian positive-definite matrix of determinant 1 over Z[o]. Then if D = 1, 3, or 7, A must be of the form B*B, where B is a matrix over Z[o]. 14. Some consequences
Some interesting number-theoretic consequences of this corollary may be derived. We take D = 3 as an example. Let p be any prime # 3. Then it is not difficult to prove that the congruence x2
+ xy + y 2 + 1
E
0 modp
always has solutions. Let (x, y ) be a solution, and set x2 1 = mp. Put ~ = [ x + c y
"'"'1
+ xy + y 2 +
m
Then A is a hermitian positive definite matrix of determinant 1 over
Z[o], and consequently must be of the form B*B, where B is a matrix over Z[w].It follows that integers a, b, c, d exist such that p
= a2
+ ab + b2 + c2 + cd + d2
The classical literature on quadratic forms is too extensive to comment on here in any detail. The results of this chapter do indicate a few of the important landmarks, however. The reader is referred to [8] or [20] for a more detailed treatment. EXERCISES AND PROBLEMS
1. Work out the consequencesof Corollary XI. 1 for the two remaining cases. 2. Let N be an arbitrary positive number. Let SN be the set of real 2 x 2 matrices whose elements do not exceed N in absolute value. Show that P2' always contains matrices which are not congruent over Z to any element of SN.