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© MAY 2020 | IRE Journals | Volume 3 Issue 11 | ISSN: 2456-8880
IRE 1702324 ICONIC RESEARCH AND ENGINEERING JOURNALS 261
On Construction of Hadamard Matrices
W. V. NISHADI1, A. A. I. PERERA2
1, 2 Department of Mathematics, Faculty of Science, University of Peradeniya, Sri Lanka
Abstract- This paper describes a specific construction
of Hadamard matrices of order with the help of
two block matrices and order under some
conditions. It is tested that this construction works
for Some related results are also given.
Indexed Terms- Block matrices, Hadamard
Matrices, Kronecker Product.
I. INTRODUCTION
Definition 1.1 A square matrix of order with
entries from the set is called
matrix if
where is the transpose of and is the identity
matrix of order .
This implies that the rows of the matrix are pairwise
orthogonal [1], [2].
It is known that Hadamard matrices exist only when
or is a multiple of 4. Surprisingly, no other
restrictions on the order of a Hadamard matrix are
known. However, the converse still remains as a
conjecture at present.
Conjecture 1.2 There exist Hadamard matrices of
order if and only if or.
A lot of research has been done on this conjecture, as
this conjecture has remained unsolved for over 100
years. Presently, the smallest order for which it is
unknown whether a real Hadamard matrix exists is
4·167 = 668. As of 2008, there are 13 multiples of 4
less than or equal to 2000 for which no Hadamard
matrix of that order is known. They are: 668, 716, 892,
1004, 1132, 1244, 1388, 1436, 1676, 1772, 1916,
1948, and 1964 [2-4].
Definition 1.3 Two Hadamard matrices of the same
order are said to be equivalent if one can be obtained
from the other by a permutation of the rows or the
columns or by multiplication of certain rows or
columns by .
Definition 1.4 A Hadamard matrix is in normal form,
or normalized, if the first row and first column of the
matrix consists of only 1’s.
Example. The followings are normalized Hadamard
matrices of order and :
(Here stands for and stands for).
II. PRELIMINARIES
Having defined Hadamard matrices and introduced
some of their most fundamental concepts, we now turn
our attention to the construction of Hadamard
matrices. Various researches have shown the
conjecture of Hadamard for special cases, though a
complete proof or counter example remains as of the
present unknown. In this section we will present four
important constructions discovered while the study of
Hadamard matrices was in its infancy.
A. Sylvester Construction
Sylvester, who was the first mathematician to consider
Hadamard matrices in 1867 and he observed that on a
chessboard, the patterns of colours in any pair of rows
agreed either everywhere or nowhere. His
investigation of the problem of constructing arrays in
which any two rows had exactly half of their entries in
common, led to the discovery of a construction method
for what would become known as Sylvester Hadamard
matrices of order [5], [6].
Theorem 1. Let be a Hadamard matrix of order .
Then the block matrix
is a Hadamard matrix of order .
© MAY 2020 | IRE Journals | Volume 3 Issue 11 | ISSN: 2456-8880
IRE 1702324 ICONIC RESEARCH AND ENGINEERING JOURNALS 262
B. Construction by Kronecker product of two
Hadamard matrices
Despite Hadamard matrices being his namesake,
Jacques Hadamard did not study Hadamard matrices
until about twenty five years after Sylvester. In fact,
Hadamard generalized Sylvester’s construction for
Hadamard matrices, as we will show shortly. For this,
we need the Kronecker product is also called Tensor
product definition [7], [8].
Definition 2.1 The Kronecker product any two
matrices and denoted by is defined
by the block matrix:
Theorem 2. If is a Hadamard matrix of order and
is a Hadamard matrix of order then is
a Hadamard matrix of order .
C. Paley Construction
Paley’s construction makes use of quadratic residues
over a field ( prime), which we introduce below.
Definition 2.2 If then is a solution
in , where .
Lemma 2.3 If , where is an odd prime, then
exactly half of the nonzero elements of are
quadratic residues.
Definition 2.4 Let be prime. The Legendre symbol
is defined to be
Theorem 3. If , where is a prime and
, then there is a Hadamard matrix of
order .
Then writing
, then is the
Hadamard matrix of order.
Theorem 4. If, where is a prime and
, then there is a Hadamard matrix of
order .
Then
is the
Hadamard matrix of order [2].
D. Williamson Construction
Williamson introduced an block matrix for his
construction:
Theorem 5. If there exist matrices and
consisting which satisfy
for every pair of distinct matrices chosen from
;
then and can be used to construct Hadamard
matrix of order [2],[9].
III. THE CONSTRUCTION
We determine the appropriate parameters of all the sub
matrices, having order that will yield a Hadamard
matrix of order We consider the following structure
of a Hadamard matrix of order and denote it
by.
(1)
To construct matrix (1), we need to find the block
matrices and of order . If
is considered
as a block matrix with sub matrices and , then the
diagonal blocks each equal . This must
be to be a Hadamard matrix. The block
is. This is going to be zero if
. Similar results hold for other off diagonal blocks.
Then we have:
© MAY 2020 | IRE Journals | Volume 3 Issue 11 | ISSN: 2456-8880
IRE 1702324 ICONIC RESEARCH AND ENGINEERING JOURNALS 263
Theorem
If and are symmetric matrices of order ,
Then in (1) is a Hadamard matrix of order if
(2)
and
(3)
Proof:
since are symmetric matrices
; since commute
and .
IV. ILLUSTRATION
In this part we construct a Hadamard matrix of order
12 with the help of method explained above.
Let
and
.
Then
,
and
.
Also, we have
.
Thus, and fulfill the requirements for to be a
Hadamard matrix.
is a Hadamard matrix of order 12.
The normalized Hadamard matrix of order 12 is:
Consider the special case of Williamson method when
. Then there exists Hadamard matrix of
order from the block matrix:
(4)
with the required conditions. Note that the block
matrices and (d) are different.
When
and
, the
normalized Hadamard matrix obtained from the
special case of Williamson’s method is given below.
© MAY 2020 | IRE Journals | Volume 3 Issue 11 | ISSN: 2456-8880
IRE 1702324 ICONIC RESEARCH AND ENGINEERING JOURNALS 264
V. CONCLUSIONS AND FUTURE WORKS
This paper progresses on the idea of construction of
Hadamard matrices in [2], [9] focusing on Williamson
construction. We introduce here block matrix of order
with the help of two other matrices of order . We
have also provided an example for the method by
constructing Hadamard matrix of order In
particular, normalized Hadamard matrix of order 12
obtained from our method and the normalized
Hadamard matrix of order 12 obtained from the
special case of Williamson method have a connection.
That is one is the transpose of the other. Notice that
block matrix (1) is not the transpose of (4).
The major question is about the existence of orders of
matrices and . This will be the concern of our
future work.
R
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