8 Further Remarks 257Hadamard’s original paper of 1893 is reproduced in [16]. Surveys on Hadamard
matrices have been given by Hedayat and Wallis [19], Seberry and Yamada [34], and
Craigen and Wallis [11]. Weighing designs are treated in Raghavarao [31]. For appli-
cations of Hadamard matrices to spectrometry, see Harwit and Sloane [18]. The proof
of Proposition 8 is due to Shahriari [35].
Our proof of Theorem 10 is a pure existence proof. A more constructive approach
was proposed by Jacobi (1846). If one applies ton×nmatrices the method which we
used for 2×2 matrices, one can annihilate a symmetric pair of off-diagonal entries. By
choosing at each step an off-diagonal pair with maximum absolute value, one obtains
a sequence of orthogonal transforms of the given symmetric matrix which converges
to a diagonal matrix.
Calculating the eigenvalues of a real symmetric matrix has important practical
applications, e.g. to problems of small oscillations in dynamical systems. House-
holder [21] and Golub and van Loan [14] give accounts of the various computational
methods available.
Gantmacher [12] and Horn and Johnson [20] give general treatments of matrix
theory, including the inequalities of Hadamard and Fischer. Our discussion of the
Hadamard determinant problem for matrices of order not divisible by 4 is mainly based
on Wojtas [37]. Further references are given in Neubauer and Ratcliffe [29].
Results of Brouwer (1983) are used in [29] to show that the upper bound in Propo-
sition 19 is attained for infinitely many values ofn. It follows that the upper bound in
Proposition 20, withm=n, is also attained for infinitely many values ofn. For if the
n×nmatrixAsatisfies
AtA=(n− 1 )In+Jn,then the 2n× 2 nmatrix
A ̄=
[
AA
A −A
]
satisfies
A ̃tA ̃=[
LO
OL
]
,
whereL= 2 AtA=( 2 n− 2 )In+ 2 Jn.
There are introductions to design theory in Ryser [33], Hall [17], and van Lint and
Wilson [25]. For more detailed information, see Brouwer [7], Lander [23] and Beth
et al.[5]. Applications of design theory are treated in Chapter XIII of [5].
We mention two interesting results which are proved in Chapter 16 of Hall [17].
Given positive integersv,k,λwithλ<k<v:
(i) Ifk(k− 1 )=λ(v− 1 )and if there exists av×vmatrixAof rational numbers
such thatAAt=(k−λ)I+λJ,thenAmay be chosen so that in additionJA=kJ.