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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