1. Introduction and Preliminaries
A weighing matrix
is a square matrix of order
n with entries 0, ±1 having
k non-zero entries per row and column and whose inner product of distinct rows (or columns) is zero. Hence,
W satisfies
. The number
k is called the
of
W. If a weighing matrix
is equal to its transpose
then
W is called
symmetric weighing matrix of order
n and weight
k. Note that
can be looked upon as the design matrix of a chemical balance weighing design. It was shown by [
1,
2] that if the variance of the errors of the weights obtained by individual weighings is
(it is assumed the balance is not biased and the errors are mutually independent and normal), then using a weighing matrix
to design an experiment to weigh
n objects will give a variance of
. Under standard linear model assumptions, following [
3], it is not hard to check that such a design is universally optimal in the class of all
n-observation chemical balance weighing designs, with
n objects, such that at most
k objects are used in each weighing. More applications of weighing matrices and further results can be found in [
4]. A great review of symmetric weighing matrices of order
n and weight
, known as symmetric conference matrices, is given by [
5]. Weighing matrices constructed from two to circulant matrices were extensively studied by the excellent review paper of [
6]. Recently, negacyclic matrices were extensively applied to construct weighing matrices in [
7]. Note that the matrices presented in that paper are not necessarily symmetric. Hence, the study of weighing matrices is important, among others, from the perspective of optimal statistical design theory. In the present paper, we propose a method to obtain families of symmetric weighing matrices.
For the construction of the new infinite classes of weighing matrices we need the following definitions and notation.
Let
be a sequence of length
n. The
non-periodic autocorrelation function (abbreviated as NPAF) of the above sequence is defined as
and the
periodic autocorrelation function (abbreviated as PAF), is defined, reducing
modulo
n, as
Circulant matrices of order
n are polynomials in the shift matrix
For example, a circulant matrix with first row is the matrix
Negacyclic matrices of order
n are polynomials in the negashift matrix
For example, a negacyclic matrix with first row is the matrix Note that the negashift matrix N is equal to the shift matrix S multiplied by a diagonal matrix of order n with at the -position and 1 elsewhere in the diagonal (i.e., ).
More details on shift matrix and the negashift matrix can be found in [
8].
Notation. We use the following notations throughout this paper.
A circulant matrix of order n with first row will be denoted as , while a negacyclic matrix of order n with first row will be denoted as .
We use to denote the monomial matrix whose elements satisfy
If
and
are two sequences of lengths
n and
m, respectively, then the
concatenate sequence C is defined as
If
is a sequence of length 1 then we write
We use − to denote and + to denote .
By we denote the transpose of the matrix D.
We use to denote the matrix with all its elements equal to 0. If no confusion is caused, a square matrix of order n with all its entries equal to zero will be denoted by .
Additional details on the above definitions and notation can be found in [
8].
In this paper, we construct two infinite families of symmetric weighing matrices. In
Section 2, a symmetric weighing matrix of order 30 and weight 25 is constructed by combining together a circulant and a negacyclic matrix with identical first row. In
Section 3, we generalise the results of
Section 2 and we present two infinite families of symmetric weighing matrices. The first family is
for all
and
while the second is
for all
and
. Open problems for symmetric weighing matrices includes the construction of symmetric weighing matrices
for a number of different values as shown in Table 2.88 of [
9]. With our infinite classes we eliminate the value
by proving and presenting existence.
2. A Symmetric Weighing Matrix of Order 30 and Weight 25
In this section, we present a method that combines a circulant and a negacyclic matrix to construct a symmetric weighing matrix. This method was applied to search for symmetric weighing matrices of order 30 and weight 25. The validity of the method is proved in Theorem 1 while the new symmetric weighing matrices are given in Corollary 1.
Theorem 1. Let be a sequence of length and entries from the set . The sequence A satisfies:
- (i)
, for all ,
- (ii)
, for all ,
and we set . Then the matrixis a symmetric weighing matrix of order and weight . Proof. To prove that D is a symmetric weighing matrix we have to show that D is a symmetric matrix and satisfy . From the definition of matrix D it is obvious that and thus D is a symmetric matrix. Moreover, due to the construction it is clear that there are exactly w non-zero elements in each row and column of D. If we show that the columns and rows are pairwise orthogonal the result will follow.
Suppose that the rows (and columns) of D are enumerated by . We shall work with rows since similar results can be derived for columns as well. We skip many pages of routine calculations for the inner product of two rows and we end up to the following results. The inner product , of two rows and with even indices difference (i.e., ) is equal to and this is equal to zero from property (ii). The inner product , of two rows and with odd indices difference (i.e., ), for some integer t, is a multiple of for some and this is equal to zero from property (i). The result follows.
The smallest example that we are able to generate from the above infinite family is given in Corollary 1. We chose this small example to be able to visualise the results and illustrate the methodology and by presenting the full matrices that are needed for the construction. The other symmetric weighing matrices that are generated from the infinite families given here, would be too big to be explicitly presented in this paper.
Corollary 1. There exist a symmetric weighing matrix of order 30 and weight 25.
Proof. Use any of the sequences given in the
Appendix A and apply Theorem 1. □
We shall illustrate the construction of a symmetric weighing matrix of order 30 and weight 25 by the next analytical example.
Example 1. We select the first sequence given in Appendix A, that is . We set as it is described in Theorem 1. Note that the symbols + and − are as denoted in the notation (point 4.). The next step is to construct the circulant and the negacyclic matrix of order 30 with first row B. We also need the monomial matrix P of order 30 which is as defined in the notation. Finally, the desirable symmetric weighing matrix of order 30 and weight 25 is . It is easy to verify that D is symmetric, satisfies and thus D is the desirable symmetric weighing matrix of order 30 and weight 25. Matrices , and D are given explicitly in the Appendix B, at the end of the paper. In the next section we give some infinite families of symmetric weighing matrices.
3. Infinite Families of Symmetric Weighing Matrices
The following Theorem gives a doubling method that retains the symmetry of the matrices.
Theorem 2. Let be a symmetric weighing matrix of order n and weight w. Then there exists a symmetric weighing matrix and for all .
Proof. With simple matrix calculations using
and
as defined above we have that
The results follow. □
Using the results of Theorem 2 we obtain the following result.
Corollary 2. Let be a symmetric weighing matrix of order n and weight w. Then there exists an infinite family of symmetric weighing matrices of order and weight for all and .
Proof. For the result follows by applying Theorem 2 with . For we apply Theorem 2 with to obtain a symmetric weighing matrix . Then we set and we apply Theorem 2 with to obtain the desirable symmetric weighing matrix .
By using the new symmetric weighing matrices of order 30 and weight 25 we present our first infinite family of symmetric weighing matrices in the next example. □
Example 2. Let be any of the new symmetric weighing matrices of order 30 and weight 25 which are constructed in Corollary 1. We use in Corollary 2 and we obtain an infinite family of symmetric weighing matrices , , .
The next Theorem doubles the order and extends the weight of a symmetric weighing matrix and also conserves the symmetry of the generated matrix.
Theorem 3. Let be a symmetric weighing matrix with zero diagonal. Then there exists an infinite family of symmetric weighing matrices for any and .
Proof. Using the given
we can construct a symmetric weighing matrix
by applying Corollary 2. The matrix
has zero diagonal and, due to the construction, the symmetric weighing matrix
also has zero diagonal. Set
The matrix
is a square matrix of order
with
non-zero elements in each row and column. We have that
and so
is a symmetric matrix. The only thing that is left to be shown is that
. We have
Thus, is the desirable symmetric weighing matrix of order and weight , for all and . □
In the next example we give a second infinite family of symmetric weighing matrices by using Theorem 3 and the new symmetric weighing matrices given in Corollary 1.
Example 3. Let be any of the new symmetric weighing matrices given in Corollary 1. We use in Theorem 3 and we obtain an infinite family of symmetric weighing matrices for all and . The proof (and the construction) follows as the proof of Theorem 3.