n-Linear Algebra of Type I and Its Applications

2008·2 MB·English

by  Vasantha Kandasamy, W. B.

n- LINEAR ALGEBRA OF TYPE I
AND ITS APPLICATIONS
W. B. Vasantha Kandasamy
e-mail:[email protected]
web: http://mat.iitm.ac.in/~wbv
www.vasantha.net
Florentin Smarandache
e-mail:[email protected]

INFOLEARNQUEST
Ann Arbor
2008
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Peer reviewers:
Professor Sukanto Bhattacharya, Queensland University,
Australia.
Dr.S.Osman, Menofia University, Shebin Elkom, Egypt.
Eng. Marian Popescu and Prof. Florentin Popescu, Craiova,
Romania.

Copyright 2008 by InfoLearnQuest and authors
Cover Design and Layout by Kama Kandasamy

Many books can be downloaded from the following
Digital Library of Science:
http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm

ISBN-10: 1-59973-074-X
ISBN-13: 978-1-59973-074-5
EAN: 9781599730745

Standard Address Number: 297-5092
Printed in the United States of America
2
CONTENTS
Preface 5
Chapter One
BASIC CONCEPTS 7

Chapter Two
n-VECTOR SPACES OF TYPE I AND THEIR
PROPERTIES 13

Chapter Three
APPLICATIONS OF
n-LINEAR ALGEBRA OF TYPE I 81

3
Chapter Four
SUGGESTED PROBLEMS 103

FURTHER READING 111

INDEX 116

ABOUT THE AUTHORS 120

4
PREFACE

With the advent of computers one needs algebraic structures
that can simultaneously work with bulk data. One such
algebraic structure namely n-linear algebras of type I are
introduced in this book and its applications to n-Markov chains
and n-Leontief models are given. These structures can be
thought of as the generalization of bilinear algebras and bivector
spaces. Several interesting n-linear algebra properties are
proved.
This book has four chapters. The first chapter just
introduces n-group which is essential for the definition of n-
vector spaces and n-linear algebras of type I. Chapter two gives
the notion of n-vector spaces and several related results which
are analogues of the classical linear algebra theorems. In case of
n-vector spaces we can define several types of linear
transformations.
The notion of n-best approximations can be used for error
correction in coding theory. The notion of n-eigen values can be
used in deterministic modal superposition principle for
undamped structures, which can find its applications in finite
element analysis of mechanical structures with uncertain
parameters. Further it is suggested that the concept of n-
matrices can be used in real world problems which adopts fuzzy
models like Fuzzy Cognitive Maps, Fuzzy Relational Equations
and Bidirectional Associative Memories. The applications of
5
these algebraic structures are given in Chapter 3. Chapter four
gives some problem to make the subject easily understandable.
The authors deeply acknowledge the unflinching support of
Dr.K.Kandasamy, Meena and Kama.

W.B.VASANTHA KANDASAMY
FLORENTIN SMARANDACHE

6
Chapter One
B C
ASIC ONCEPTS

In this chapter we introduce the notion of n-field, n-groups (n (cid:2)
2) and illustrate them by examples. Throughout this book F will
denote a field, Q the field of rationals, R the field of reals, C the
field of complex numbers and Z , p a prime, the finite field of
p
characteristic p. The fields Q, R and C are fields of zero
characteristic.

Now we proceed on to define the concept of n-groups.

DEFINITION 1.1: Let G = G (cid:3) G (cid:3) … (cid:3) G (n (cid:2) 2) where
1 2 n
(cid:4)
each (G, *, e) is a group with the binary operation and e
i i i i i
the identity element, such that G (cid:5) G, if i (cid:5) j, 1 (cid:2) j, i (cid:2) n.
i j
Further G (cid:6) G or G (cid:6) G if i (cid:5) j. Any element x (cid:7) G would be
i j j i
represented as x = x (cid:3) x (cid:3) …(cid:3) x ; where x (cid:7) G, i = 1, 2, …,
1 2 n i i
n. Now the operations on G is described so that G becomes a
group. For x, y (cid:7) G, where x = x (cid:3) x (cid:3) …(cid:3) x and y = y (cid:3) y
1 2 n 1 2
(cid:3) … (cid:3) y ; with x, y (cid:7) G, i = 1, 2, …, n.
n i i i
x * y = (x (cid:3) x (cid:3) …(cid:3) x ) * (y (cid:3) y (cid:3) … (cid:3) y )
1 2 n 1 2 n
= (x * y (cid:3) x * y (cid:3) … (cid:3) x * y ).
1 1 1 2 2 2 n n n

7
Since each x * y (cid:7) G we see x* y = (p (cid:3) p (cid:3) …(cid:3) p ) where
i i i i 1 2 n
x * y = p for i = 1, 2, …, n. Thus G is closed under the binary
i i i i
operation *.
Now let e = (e (cid:3) e (cid:3) … (cid:3) e ) where e (cid:7) G the identity of
1 2 n i i
G with respect to the binary operation, *, i = 1, 2, …, n we see
i i
e * x = x * e = x for all x (cid:7) G. e will be known as the identity
element of G under the operation *.
Further for every x = x (cid:3) x (cid:3) … (cid:3) x (cid:7) G; we have
1 2 n
x(cid:8)1(cid:3)x(cid:8)1(cid:3)...(cid:3)x(cid:8)1 in G such that,
1 2 n
x*x(cid:8)1 (cid:9)(x (cid:3)x (cid:3)...(cid:3)x )*(x(cid:8)1(cid:3)x (cid:8)1(cid:3)...(cid:3)x (cid:8)1)
1 2 n 1 2 n
= x * x(cid:8)1 (cid:3)x * x (cid:8)1(cid:3)...(cid:3)x * x (cid:8)1
1 1 1 2 2 2 n n n
= x-1 * x
(e (cid:3) e (cid:3) … (cid:3) e ) = e.
1 2 n
x(cid:8)1 (cid:9)x(cid:8)1(cid:3)x (cid:8)1(cid:3)...(cid:3)x (cid:8)1
1 2 n

is known as the inverse of x = x (cid:3) x (cid:3) … (cid:3) x . We define (G,
1 2 n
*, e) to be the n-group (n (cid:2) 2). When n = 1 we see it is the
group. n = 2 gives us the bigroup described in [37-38] when n
> 2 we have the n-group.

Now we illustrate this by examples before we proceed on to
recall more properties about them.

Example 1.1: Let G = G (cid:3) G (cid:3) G (cid:3) G (cid:3) G where G = S
1 2 3 4 5 1 3
the symmetric group of degree 3 with

(cid:10)1 2 3(cid:11)
e (cid:9)(cid:12) (cid:13),
1 (cid:14)1 2 3(cid:15)

G = (cid:16)g | g6 = e (cid:17), the cyclic group of order 6, G = Z , the group
2 2 3 5
under addition modulo 5 with e = 0, G = D = {a, b | a2 = b8 =
3 4 8
1; bab = a}, the dihedral group of order 8, e = 1 is the identity
4
element of G and G = A the alternating subgroup of S with
4 5 4 4

(cid:10)1 2 3 4(cid:11)
e (cid:9)(cid:12) (cid:13).
4 (cid:14)1 2 3 4(cid:15)
8
Clearly G = S (cid:3) G (cid:3) Z (cid:3) D (cid:3) A is a n-group with n = 5.
3 2 5 8 4

Any x (cid:7) G would be of the form

(cid:10)1 2 3(cid:11) (cid:10)1 2 3 4(cid:11)
x(cid:9)(cid:12) (cid:13)(cid:3)g2 (cid:3)4(cid:3)b3(cid:3)(cid:12) (cid:13).
(cid:14)2 1 3(cid:15) (cid:14)1 3 4 2(cid:15)

(cid:10)1 2 3(cid:11) (cid:10)1 2 3 4(cid:11)
x-1 = (cid:12) (cid:13)(cid:3)g4 (cid:3)1(cid:3)b5(cid:3)(cid:12) (cid:13).
(cid:14)2 1 3(cid:15) (cid:14)1 4 2 3(cid:15)

The identity element of G is

(cid:10)1 2 3(cid:11) (cid:10)1 2 3 4(cid:11)
(cid:12) (cid:13)(cid:3)e (cid:3)0(cid:3)1(cid:3)(cid:12) (cid:13)
(cid:14)1 2 3(cid:15) 2 (cid:14)1 2 3 4(cid:15)
= e (cid:3) e (cid:3) e (cid:3) e (cid:3) e .
1 2 3 4 5

Thus G is a 5-group. Clearly the order of G is o(G ) (cid:18) o(G ) (cid:18)
1 2
o(G ) (cid:18) o(G ) (cid:18) o(G ) = 6 (cid:18) 6 (cid:18) 5 (cid:18) 16 (cid:18) 12 = 34, 560.
3 4 5
We see o(G) < (cid:19). Thus if in the n-group G (cid:3) G (cid:3) … (cid:3)
1 2
G , every group G is of finite order then G is of finite order; 1 (cid:20)
n i
i (cid:20) n.

Example 1.2: Let G = G (cid:3) G (cid:3) G where G = Z , the group
1 2 3 1 10
under addition modulo 10, G = (cid:16)g | g5 = 1(cid:17), the cyclic group of
2
order 5 and G = Z the set of integers under +.
3
Clearly G is a 3-group. We see G is an infinite group for
order of G is infinite.
3
Further it is interesting to observe that every group in the 3-
group G is abelian. Thus if G = G (cid:3) G (cid:3) … (cid:3) G , is a n-
1 2 n
group (n (cid:2) 2), we see G is an abelian n-group if each G is an
i
abelian group; i = 1, 2, …, n. Even if one of the G in G is a non
i
abelian group then we call G to be only a non abelian n-group.

Having seen an example of an abelian and non abelian group we
now proceed on to define the notion of n-subgroup. We need all
9