Number Theory: An Introduction to Mathematics

2 Hadamard Matrices 229

detA=det(SA)=detDδ

and

det(AB)=det(SAB)=det(DδB).

But detDδ =δ,by(vi),anddet(DδB)=δdetB, by (ii). Therefore det(AB)=
detA·detB.
This completes the proof of existence.

Corollary 2If A∈Mnand if Atis the transpose of A, thendetAt=detA.

Proof The mapd: Mn → Fdefined byd(A)=detAtalso has the properties
(i)′, (ii)′.

The proof of Theorem 1 shows further thatSLn(F)is the special linear group,
consisting of all A∈MnwithdetA=1.
We do not propose to establish here all the properties of determinants which we
may later require. However, we note that if

A=

[

B 0

CD

]

is a partitioned matrix, whereBandDare square matrices of smaller size, then

detA=detB·detD.

It follows that ifA=(αjk)islower triangular(i.e.αjk=0forallj,kwithj<k)
orupper triangular(i.e.αjk=0forallj,kwithj>k), then

detA=α 11 α 22 ···αnn.

2 HadamardMatrices..........................................

We begin by obtaining an upper bound for det(AtA),whereAis ann×mreal matrix. If
m=n,thendet(AtA)=(detA)^2 and bounding det(AtA)is the same as Hadamard’s
problem of bounding|detA|.However,aswewillseein§3, the problem is of interest
also form<n.
In the statement of the following result we denote by‖v‖the Euclidean norm of
a vectorv =(α 1 ,...,αn)∈Rn. Thus‖v‖≥0and‖v‖^2 =α^21 +···+αn^2 .The
geometrical interpretation of the result is that a parallelotope with given side lengths
has maximum volume when the sides are orthogonal.

Proposition 3Let A be an n×m real matrix with linearly independent columns
v 1 ,...,vm.Then

det(AtA)≤

∏m

k= 1

‖vk‖^2 ,

with equality if and only if AtA is a diagonal matrix.