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

⎞

⎟

⎟

⎟

⎟

.