4 1. FROM QUADRATIC FORMS TO SPHERE PACKINGS AND COVERINGS
2q12
q11
q22
2q12
trace Q = 4
q22 q11
2 1
1 2
2 0
0 2
0 0
0 4
2 −2
−2 2
Figure 2. Binary PQFs form a convex cone in R3. PQFs with
constant trace lie in an open circle; its boundary contains positive
semidefinite forms.
A quadratic form, respectively real symmetric matrix Q
Sd,
is positive semi-
definite if Q[x] 0 for all x
Rd
\ {0}. The set of positive semidefinite forms,
denoted by S≥0,
d
is the closure of
Sd
0
. The group GLd(Z) acts on S≥0
d
and
Sd
0
by
Q Q[U] = U
tQU.
Recall from linear algebra that the same group action on
Sd
0
by
GLd(R) = A
Rd×d
: det A = 0
has only one orbit, that is, for every Q
Sd
0
there exists a linear substitution
S GLd(R) with
StQS
= idd. We refer to the recent article [123] of Gruber for
more information on the rich structure of the convex cones
Sd
0
and
S≥0.d
1.1.5. Decompositions. It is well known that every PQF Q can be written
as a sum of squares. For example, the Lagrange expansion of Q is given by
(1.7) Q[x] =
d
i=1
Ai

⎝xi

d
j=i+1
αij xj
⎞2

,
with unique positive outer coefficients Ai and inner coefficients αij R, for i =
1, . . . , d and j = i + 1, . . . , d. The inner and outer coefficients provide another
parameter space for PQFs. Written as a matrix equation, we obtain the Jacobi
identity (Iwasawa decomposition)
(1.8) Q =
StDS
with diagonal matrix D and upper triangular matrix S with diagonal entries equal
to 1:
(1.9) D =






A1 0 . . . 0
0
..
.
.
..
.
.
.
.
.
.
..
.
.
..
0
0 . . . 0 Ad






, S =



⎜0


1 −α12 . . . −α1d
..
.
..
.
.
.
.
.
.
.
..
.
..
.
−αd−1,d⎠
0 . . . 0 1





.
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