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Book ChapterDOI

A construction for {0,1,-1} orthogonal matrices visualized

TL;DR: Propus is a construction for orthogonal \(\pm 1\) matrices, which is based on a variation of the Williamson array, called the propus array.
Abstract: Propus is a construction for orthogonal \(\pm 1\) matrices, which is based on a variation of the Williamson array, called the propus array $$\begin{aligned} \left[ \begin{matrix} A&{} B &{} B &{} D \\ B&{} D &{} -A &{}-B \\ B&{} -A &{} -D &{} B \\ D&{} -B &{} B &{}-A \end{matrix} \right] . \end{aligned}$$

Summary (1 min read)

1 Introduction

  • Hadamard matrices arise in statistics, signal processing, masking, compression, combinatorics, error correction, coil winding, weaving, spectroscopy and other areas.
  • The authors show how finding propus-Hadamard matrices using Williamson matrices and D-optimal designs can be easily seen through their pictures.

1.1 Definitions and Basics

  • Symmetric Hadamard matrices made using propus like matrices will be called symmetric propus-Hadamard matrices.
  • The authors define the following classes of propus like matrices.
  • The authors note that there are slight variations in the matrices which allow variant arrays and non-circulant matrices to be used to give symmetric Hadamard matrices,.
  • These can also be used with generalized matrices ([33]).
  • Symmetric Hadamard matrices made using propus like matrices will be called symmetric propus-Hadamard matrices.

2.3 Order 4n from Williamson Matrices using q a Prime Power

  • Some of these cases arise when q is a prime power, however the DelsarteGoethals-Seidel-Turyn construction means the required circulant matrices also exist for these prime powers .
  • Then propus-Hadamard matrices exist for order 4n.
  • The authors are interested in those cases where the D-optimal design is constructed from two circulant matrices one of which must be symmetric.
  • The authors see clearly, looking first at GP28 in Figure 6 where the D-optimal design is highlighted in purple, the construction method.
  • Now the method will also be clear in GP12 and GP76.

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University of Wollongong University of Wollongong
Research Online Research Online
Faculty of Engineering and Information
Sciences - Papers: Part B
Faculty of Engineering and Information
Sciences
2018
A construction for {0,1,-1} orthogonal matrices visualized A construction for {0,1,-1} orthogonal matrices visualized
N A. Balonin
Saint-Petersburg State University of Aerospace Instrumentation
, korbendfs@mail.ru
Jennifer Seberry
University of Wollongong
, jennie@uow.edu.au
Follow this and additional works at: https://ro.uow.edu.au/eispapers1
Part of the Engineering Commons, and the Science and Technology Studies Commons
Recommended Citation Recommended Citation
Balonin, N A. and Seberry, Jennifer, "A construction for {0,1,-1} orthogonal matrices visualized" (2018).
Faculty of Engineering and Information Sciences - Papers: Part B
. 1371.
https://ro.uow.edu.au/eispapers1/1371
Research Online is the open access institutional repository for the University of Wollongong. For further information
contact the UOW Library: research-pubs@uow.edu.au

A construction for {0,1,-1} orthogonal matrices visualized A construction for {0,1,-1} orthogonal matrices visualized
Disciplines Disciplines
Engineering | Science and Technology Studies
Publication Details Publication Details
Balonin, N. & Seberry, J. (2018). A construction for {0,1,-1} orthogonal matrices visualized. Lecture Notes
in Computer Science, 10765 47-57. Newcastle 17-21 July, 2017 28th International Workshop, IWOCA 2017
This journal article is available at Research Online: https://ro.uow.edu.au/eispapers1/1371

A Construction for {0,1,-1} Orthogonal Matrices
Visualized
N. A. Balonin
and Jennifer Seberry
Dedicated to the Unforgettable Mirka Miller
Abstract
Propus is a construction for orthogonal
±
1 matrices, which is based
on a variation of the Williamson array, called the propus array
A B B D
B D A B
B A D B
D B B A
.
This array showed how a picture made is easy to see the construction
method. We have explored further how a picture is worth ten thousand
words.
We give variations of the above array to allow for more general
matrices than symmetric Williamson propus matrices. One such is the
Generalized Propus Array (GP).
Keywords: Hadamard Matrices,
D
-optimal designs, conference matrices,
propus construction, Williamson matrices; visualization; 05B20.
1 Introduction
Hadamard matrices arise in statistics, signal processing, masking, compres-
sion, combinatorics, error correction, coil winding, weaving, spectroscopy
and other areas. They been studied extensively. Hadamard showed [
14
]
the order of an Hadamard matrix must be 1, 2 or a multiple of 4. Many
constructions for
±
1 matrices and similar matrices such as Hadamard ma-
trices, weighing matrices, conference matrices and
D
-optimal designs use
skew and symmetric Hadamard matrices in their construction. For more
details see Seberry and Yamada [
30
]. Different constructions are most useful
Saint Petersburg State University of Aerospace Instrumentation, 67, B. Morskaia St.,
190000, St. Petersburg, Russian Federation. Email: korbendfs@mail.ru
Scho ol of Computing and Information Technology, EIS, University of Wollongong,
NSW 2522, Australia. Email: jennifer seberry@uow.edu.au
1

in different cases. For example the Paley I construction for spectroscopy and
the Sylvester construction for Walsh functions (discrete Fourier transforms)
for signal processing.
An Hadamard matrix of order
n
is an
n ×n
matrix with elements
±
1
such that
HH
=H
H =nI
n
, where
I
n
is the
n ×n
identity matrix and
stands for transposition. A skew Hadamard matrix
H =I +S
has
S
=S
.
For more details see the books and surveys of Jennifer Seberry (Wallis) and
others [30, 34] cited in the bibliography.
Propus is a construction method for symmetric orthogonal
±
1 matrices,
using four matrices A, B =C, and D, where
AA
+2BB
+DD
= constant I,
based on the array
A B B D
B D A B
B A D B
D B B A
.
It gives aesthetically pleasing visual images (pictures) when converted
using MATLAB (we show some below).
We show how finding propus-Hadamard matrices using Williamson matri-
ces and
D
-optimal designs can be easily seen through their pictures. These
can be generalized to allow non-circulant and/or non-symmetric matrices
with the same aim to give symmetric Hadamard matrices.
We illustrate two constructions to show the construction method (these
are proved in [2])
q
1
(mod
4
)
, a prime power, such matrices exist for order
t =
1
2
(q +
1
)
,
and thus propus-Hadamard matrices of order 2
(q +
1
)
(this uses the
Paley II construction) ;
t
3
(mod
4
)
, a prime, such that
D
-optimal designs, constructed using
two circulant matrices, one of which must be circulant and symmetric,
exist of order 2
t
, then such propus-Hadamard matrices exist for order
4t.
We note that appropriate Williamson type matrices may also be used to
give propus-Hadamard matrices but do not pursue this avenue in this paper.
There is also the possibility that this propus construction may lead to some
insight into the existence or non-existence of symmetric conference matrices
for some orders. We refer the interested reader to mathscinet.ru/catalogue/
propus/.
2

1.1 Definitions and Basics
Two matrices X and Y of order n are said to be amicable if XY
=Y X
.
A
D
-optimal design of order 2
n
is formed from two commuting or amicable
(
±
1) matrices,
A
and
B
, satisfying
AA
+BB
=(
2
n
2
)I +
2
J
,
J
the matrix
of all ones, written in the form
DC =[
A B
B
A
] and DA =[
A B
B A
].
respectively. In figure 1 the structure is clear to see.
(a) D6 (n = 3) (b) D14 (n = 7) (c) D38 (n = 19)
Figure 1: D-optimal designs for orders 2n
Symmetric Hadamard matrices made using propus like matrices will be
called symmetric propus-Hadamard matrices.
We define the following classes of propus like matrices. We note that
there are slight variations in the matrices which allow variant arrays and
non-circulant matrices to be used to give symmetric Hadamard matrices,
All propus like matrices
A
,
B =C
,
D
are
±
1 matrices of order
n
satisfy the
additive property
AA
+2BB
+DD
=4nI
n
. (1)
We make the definitions following [2]:
propus matrices: four circulant symmetric
±
1 matrices,
A
,
B
,
B
,
D
of
order n, satisfying the additive property (use P );
propus-type matrices: four pairwise amicable
±
1 matrices,
A
,
B
,
B
,
D
of order n, A
=A, satisfying the additive property (use P );
generalized-propus matrices: four pairwise commutative
±
1 matrices,
A
,
B
,
B
,
D
of order
n
,
A
=A
, which satisfy the additive property
(use GP ).
We use two types of arrays into which to plug the propus like matrices:
the Propus array,
P
, or the generalized-propus array,
GP
. These can also be
used with generalized matrices ([33]).
3

Citations
More filters
Journal ArticleDOI
TL;DR: A relative bound for the number of lines in a Euclidean space that are pairwise orthogonal or at fixed angle is derived and all such graphs whose negative eigenvalue in not less than − 2 are determined, except for so-called exceptional signed graphs.

19 citations

References
More filters
Journal Article
TL;DR: The On-Line Encyclopedia of Integer Sequences (OEIS) as mentioned in this paper is a database of 13,000 number sequences and is freely available on the Web (http://www.att.com/~njas/sequences/) and is widely used.
Abstract: The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences. It is freely available on the Web (http://www.research.att.com/~njas/sequences/) and is widely used. There are several ways in which it benefits research: 1 It serves as a dictionary, to tell the user what is known about a particular sequence. There are hundreds of papers which thank the OEIS for assistance in this way. 1 The associated Sequence Fans mailing list is a worldwide network which has evolved into a powerful machine for tackling new problems. 1 As a direct source of new theorems, when a sequence arises in two different contexts. 1 As a source of new research, when one sees a sequence in the OEIS that cries out to be analyzed. The 40-year history of the OEIS recapitulates the story of modern computing, from punched cards to the internet. The talk will be illustrated with numerous examples, emphasizing new sequences that have arrived in the past few months. Many open problems will be mentioned. Because of the profusion of books and journals, volunteers play an important role in maintaining the database. If you come across an interesting number sequence in a book, journal or web site, please send it and the reference to the OEIS. (You do not need to be the author of the sequence to do this.) There is a web site for sending in "Comments" or "New sequences". Several new features have been added to the OEIS in the past year. Thanks to the work of Russ Cox, searches are now performed at high speed, and thanks to the work of Debby Swayne, there is a button which displays plots of each sequence. Finally, a "listen" button enables one to hear the sequence played on a musical instrument (try Recamaan's sequence A005132!).

4,548 citations

Book ChapterDOI
Neil J. A. Sloane1
27 Jun 2007
TL;DR: The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences which serves as a dictionary, to tell the user what is known about a particular sequence and is widely used.
Abstract: The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences. It is freely available on the Web (http://www.research.att.com/~njas/sequences/) and is widely used. There are several ways in which it benefits research: 1 It serves as a dictionary, to tell the user what is known about a particular sequence. There are hundreds of papers which thank the OEIS for assistance in this way. 1 The associated Sequence Fans mailing list is a worldwide network which has evolved into a powerful machine for tackling new problems. 1 As a direct source of new theorems, when a sequence arises in two different contexts. 1 As a source of new research, when one sees a sequence in the OEIS that cries out to be analyzed. The 40-year history of the OEIS recapitulates the story of modern computing, from punched cards to the internet. The talk will be illustrated with numerous examples, emphasizing new sequences that have arrived in the past few months. Many open problems will be mentioned. Because of the profusion of books and journals, volunteers play an important role in maintaining the database. If you come across an interesting number sequence in a book, journal or web site, please send it and the reference to the OEIS. (You do not need to be the author of the sequence to do this.) There is a web site for sending in "Comments" or "New sequences". Several new features have been added to the OEIS in the past year. Thanks to the work of Russ Cox, searches are now performed at high speed, and thanks to the work of Debby Swayne, there is a button which displays plots of each sequence. Finally, a "listen" button enables one to hear the sequence played on a musical instrument (try Recamaan's sequence A005132!).

3,347 citations

Book
01 Jan 1992
TL;DR: In this article, the authors presented a model for orthogonal factorizations of complete graphs, including room square and block design, and showed how to construct room squares and block designs.
Abstract: Orthogonal Factorizations of Graphs (B. Alspach, et al.). Conjugate--Orthogonal Latin Squares and Related Structures (F. Bennett & L. Zhu). Directed and Mendelsohn Triple Systems (C. Colbourn & A. Rosa). Room Squares and Related Designs (J. Dinitz & D. Stinson). Steiner Quadruple Systems (A. Hartman & K. Phelps). Difference Sets (D. Jungnickel). Decomposition Into Cycles II: Cycle Systems (C. Lindner & C. Rodger). Coverings and Packings (W. Mills & R. Mullin). Colorings of Block Designs (A. Rosa & C. Colbourn). Hadamard Matrices, Sequences, and Block Designs (J. Seberry & M. Yamada). Large Sets of Disjoint Designs and Related Structures (L. Teirlinck). One--Factorizations of Complete Graphs (W. Wallis). Index.

511 citations

Frequently Asked Questions (2)
Q1. What have the authors contributed in "A construction for {0,1,-1} orthogonal matrices visualized" ?

In this paper, Seberry and Yamada showed how to construct symmetric Hadamard matrices using the generalized propus array. 

There are many constructions and variations of the propus theme to be explored in future research. There is the possibility that these visualizations may be used for quilting.