This paper presents an up-to-date review of the developments made in the field of rhotrix theory for a decade, starting from the year 2003, when the concept of rhotrix was introduced, up to the end of 2013. Over forty articles on rhotrix theory have been published in journals since its inception, indicating the need for a first review.

Rhotrix Matrix Rhotrix Theory Matrix Theory
1. Introduction

In the year 2003, a relatively new paradigm of science, now known as rhotrix theory was initiated by Ajibade  , as an extension of ideas, on matrix-tertions and matrix-noitrets proposed by Atanassov and Shannon  . Since the publication of the article titled as “the concept of rhotrix for mathematical enrichment” in  , many researchers have shown interest in the improvement of the theories and applications of rhotrices for the past one decade.

In the literature of rhotrix theory, starting from 2003, over forty articles have been published, thereby requiring the need for a first review. Before going further, it is pertinent to mention that two methods for multiplication of rhotrices having the same size are currently available in literature. The first one is “the heart based method for rhotrix multiplication” defined in  , where the initial algebra and analysis of rhotrices were presented. The second alternative method is the row-column based method for rhotrix multiplication proposed by Sani   , in an attempt to answer the question of “finding a transformation for conversion rhotrix to matrix and vice versa” posed by Ajibade in the concluding section of his article. However, each method provides enabling environment to explore the usefulness of rhotrices as tools for carrying out mathematical research.

The objective of this article is to give a comprehensive literature survey of all published articles on rhotrix theory, since the introduction of the concept in 2003, up to the end of 2013. To achieve this, we classify all the over fourty articles in the literature of rhotrix theory into two classes. We term one class of the articles in the literature of rhotrix theory as commutative rhotrix theory, while the other class as non-commutative rhotrix theory. The reason behind this classification is due to the fact that, contributory author(s) of a single article on rhotrix theory adopted either Ajibade’s heart-based method for multiplication of rhotrices or Sani’s row-column me- thod for multiplication of rhotrices in carrying out the work.

The choice of the two class names: commutative rhotrix theory and non-commutative rhotrix theory arise, respectively, from the commutative property inherent with the heart-based method for rhotrix multiplication, and the non-commutative property associated with row-column based method for rhotrix multiplication.

In line with this, articles on rhotrix theory can be broadly categorized according to the method of rhotrix multiplication used in presenting the work as follows:

1) Commutative rhotrix theory, i.e. Ajibade’s article and all other articles using the Ajibade’s heart-based method for rhotrix multiplication.

2) Non-commutative rhotrix theory, i.e. singularly authored articles by Sani and all other articles using Sani’s row-column based method for rhotrix multiplication.

This survey paper contains three other sections after the introductory section. Section 2 presents the survey of developments in rhotrix theory. Section 3 analyzes these developments and then Section 4 presents the conclusion.

2. Survey of Developments on Rhotrix Theory

This section presents a review of developments on rhotrix theory in a systematize form, starting with the review of commutative rhotrix theory in Subsection 2.1 and then followed by the review of non-commutative rhotrix theory in Subsection 2.2.

2.1. Class of Commutative Rhotrix Theory

Table 1 illustrates the title list of all journal articles that used Ajibade’s heart based method for rhotrix multiplication, available in the literature of rhotrix theory, starting from 2003 and to the end of 2013. Thus, articles in Table 1 belong to the class of commutative rhotrix theory. Now, we start a systemic review of these works in Table 1 as follows:

In  Ajibade introduced the concept of rhotrix of size three as

where is called the heart of any rhotrix. The operations of addition, scalar multiplication and multiplication are defined for rhotrices of size three in  . These rhotrix operations defined for rhotrix set of size three in  were thereafter, extended to rhotrix set of size in the Ph.D. thesis of Mohammed  , and recorded as follows: let

List of titles in journals published from 2003 to 2013 that belong to the class of commutative rhotrix theory
S/no.Title
1A note on the rhotrix system of equations
2A note on rhotrix exponent rule and its applications to special series and polynomial equation defined over rhotrices
3A remark on the classifications of rhotrices as abstract structures
4Algebraic properties of singleton, coiled and modulo rhotrices
5Certain field of fractions
7Enrichment exercises through extension to rhotrices
8Generalization and algorithmatization of heart based method for multiplication of rhotrices
9Note on certain field of fractions
10Note on rhotrices and the construction of finite fields
11On construction of rhomtrees as graphical representation of rhotrices
12On the structure of rhotrix
13On the linear system over rhotrices
14Rhotrices and the construction of finite fields
15Rhotrix polynomials and polynomial rhotrices
16Rhotrix sets and rhotrix spaces category
17Rhotrix topological spaces
18The concept of rhotrix in mathematical enrichment
19The concept of heart oriented rhotrix multiplication

be the set of all real rhotrices of size, where, , is the integer value obtained

on division of by 2, and is the heart of any rhotrix in. Let and be

any two rhotrices in and scalar. Then

(4)

where and are the hearts of rhotrices and respectively.

The extended rhotrix multiplication (4) was named in  as “Ajibade’s heart-based method for multiplication of rhotrices”. The rhotrix operations defined in  was adopted by  to present various classifications of rhotrices and their expressions as abstract structures of groups, semigroups, monoids, rings and Boolean algebras. The theorem for rhotrix exponent rule was first proposed without proof in  , thereafter,  established and characterized the theorem for rhotrix exponent rule and extended the result to systemization of expressing special series and polynomial equations over rhotrices.

A remark on classifications of rhotrices as abstract structures was proposed by  over rhotrices. In the work, rhotrix ring was characterized; rhotrix integral domain and rhotrix field were constructed with certain conditions. Construction of certain field of fractions over rhotrices was presented by  as an extension to  . It was made known in  that the rhotrix field in   holds only if the set of all hearty rhotrices of size three given in  is used as the underlying set.

The generalization of Ajibade’s heart based method for rhotrix multiplication in  was algorithmatized for computing machines by  . A simplification of rhotrix expression generalization in  was presented by  . Construction and analysis of metric topological spaces using rhotrix set as the underlying set were considered in  .

The concept of tree in graph theory was extended to rhotrix theory by  through their introduction of

rhomtrees of order as graphical representation of rhotrices of size, where. It

was shown in their work that these rhomtrees have connection to known real world models such as topology of computing network, methane compound and certain product of sets.

In  , the algebraic properties of singleton, coiled and modulo rhotrices were presented. Investigations of various constructions of finite fields over rhotrices were carried out in both   . The cardinality of these finite fields was calculated through concrete examples. A study of the structure of rhotrices having entries from the set of integers modulo and their properties was conducted by  . The rhotrix quadratic polynomial presented as part of a note on rhotrix exponent rule and its applications in  was given certain extensions by  . Rhotrix polynomial and its extension to construction of rhotrix polynomial ring was proposed in  . An investigation of rhotrix sets and rhotrix spaces categorized over numbers in real and complex fields was presented by  . A system of linear equations arising from the rhotrix equation was investigated in  and the conditions for their solvability were determined in the article. A note on rhotrix system of equations was presented by  as an extension to earlier work considered in  . The system of rhotrix equations was solved simultaneously.

2.2. Class of Non-Commutative Rhotrix Theory

Table 2 illustrates the title list of all journal articles that used row-column based method for rhotrix multiplication, available in the literature of rhotrix theory from 2003 to 2013. Thus, articles in Table 2 belong to the class of non-commutative rhotrix theory. Now, we start a systemic review of works in Table 2 as follows:

Sani  proposed (5) as an alternative method for multiplication of rhotrices of size three as an attempt to answer the question of “how can one convert a rhotrix to matrix and then vice versa”, posed in the concluding

List of titles published from 2003 to 2013 that belong to the class of non-commutative rhotrix theory
S/no.Titles
1A determinant method for solving rhotrix system of eqn.
2A note on relationship between invertible rhotrices and associated invertible matrices
3Adjacent rhotrix of a complete, simple and undirected graph
4Adjoint of a rhotrix and its basic properties
5Algorithm design for row-column multiplication of n-dimensional rhotrices
6An alternative method for multiplication of rhotrices
7An example of linear mappings: extension to rhotrices
8Cayley-Hamilton theorem in rhotrix
9Conversion of a rhotrix to a coupled matrix
10Hilbert matrix and its relationship with a special rhotrix
11On inner product space and bilinear forms over rhotrices
12On involutory and Pascal rhotrices
13On the construction of involutory rhotrices.
14Parallel multiplication of rhotrices using systolic array architecture
15Rhotrix multiplication on two-dimensional process grid topologies
16Rhotrices and elementary row operations
17Rhotrix linear transformation
18Rhotrix vector spaces
19Row-wise representation of arbitrary rhotrix
20Solution of two coupled matrices
21The Cayley-Hamilton theorem for rhotrices
22The equation, over rhotrices
23The row-column multiplication of high dimensional rhotrices

section in  .

This multiplication was later generalized by  to multiplication of rhotrices of size n as:

where and.

This (6) was presented in  as extended row-column based method for rhotrix multiplication.

In  , a presentation of the concept of Hilbert rhotrix and its relationship with well known Hilbert matrix was done. A special rhotrix termed as “Hilbert rhotrix” of size 5 was shown as a couple of two Hilbert matrices of sizes and.

A method of converting rhotrix to a special form of matrix called “coupled matrix” was given in  . This was achieved through rotating the rhotrix of size at an angle of 45˚ in anti-clockwise direction, which result into a special form of matrix with missing values. For example, a rhotrix of size 5 can express as a coupled matrix through half transpose as follows:

where indicates a rotation through 45˚ in anti-clockwise direction. The special matrix in (7) is a coupling of matrix with a matrix, hence, the name “coupled matrix”. Thus, in general,

This is a rhotrix expressed as a coupled matrix of dimension, coupling a matrix with a

matrix, where.

Two coupled matrices and can be multiplied together by simply filling the missing spaces with zeros, after the multiplication, we removed the zero in other to have the result in filled coupled matrix form. The following is a very useful result recorded from Sani  .

2.3. Theorem

If a coupled matrix is completed with zeros, then its determinants is the product of the determinants of

the matrices and, where.

This result on coupled matrix is very significant because it can be applied to solve problems involving two different systems of linear equations simultaneously, where one is a system, while the other is a system,.

Sani  presented the solution of two coupled matrices by extending the idea of a coupled matrix in  to a general case involving and matrices.

A one-sided system of the form, where, is an n-dimensional rhotrix, the unknown n-dimensional rhotrix vector and the right hand side rhotrix vector was presented by  . The necessary and sufficient conditions for the solvability of the system of an n-dimensional rhotrix equation were discussed. Furthermore, the eigenvalues and the corresponding eigenvectors problems were solved.

The rhotrix addition and scalar multiplication defined in  was expressed in form of coupled matrices in  . The ideas were used by  to generalize and characterize the rhotrix vector space of size 3 initiated in  to rhotrix vector space of size n, through expression of rhotrices as coupled matrices.

Following this,  presented the concept of linear mapping to rhotrices and present its properties. It was shown in the work that the proposed method of converting a rhotrix to a “coupled matrix” (8) as defined in  is also a linear mapping.

In  an algorithm design for Sani’s row-column based method for rhotrix multiplication (6) was proposed. As an extension to    , various method of representing an arbitrary rhotrix was identified by  . One of the methods is the row-wise method, observed in the article to be flexible in analyzing rhotrices for mathematical enrichment. The flexibility of the representation has paved way for two formulae, one for row-column based method for arbitrary rhotrix multiplication and the other for heart-based method for arbitrary rhotrix multiplication.

The Cayley-Hamilton theorem for matrix is one of the well-known results in linear algebra. In 2012, the equivalence of this result was considered for rhotrix Cayley-Hamilton theorem in both   . A note on relationship between invertible rhotrices and associated invertible matrices was proposed by  . A study of adjoint of a rhotrix and its basic properties was presented by  . The concept of inner product and bilinear forms over real rhotrices was considered in  . A determinant method for solving rhotrix system of linear equations was presented by  .

It is well known that an involutory matrix is a matrix that is its own inverse. Such matrices are of great im- portance in matrix theory and algebraic cryptography. In  a method for constructing involutory rhotrices and their properties was given. Thereafter, an extension to  was given by  through the development of some

Number and percentage of articles on rhotrix theory per rhotrix theory class
S/no.CategoryPapersPercentage (%)
1Class of commutative rhotrix theory1945.24
2Class of non-commutative rhotrix theory2354.46
Total42100

theorems on involution in the context of rhotrices. Also, the description of Pascal rhotrices and their related properties was also considered. In  , the theory of graph was extended to consider adjacent rhotrix of a complete, simple and undirected graph. A consideration of parallel multiplication of rhotrices using systolic array architecture was presented by  . Thereafter, a rhotrix multiplication on two-dimensional process grid topologies was carried out by  . The concept of rhotrix linear transformation with a number of theorems was presented by  . An investigation of rhotrices and its elementary row operations was carried out in  .

3. Analysis

In this section, we present two tables for the analysis of articles in the literature review of rhotrix theory. In Table 3, we specify the number and percentage of articles on rhotrix theory from 2003 to the end of 2013 per rhotrix theory class.

The remarkable aspect of this literature review of articles on rhotrix theory is that authors following the class of commutative rhotrix theory enjoy the commutative property associated with the heart based method for rhotrix multiplication. For this reason, a number of abstract structures such as rhotrix groups, rhotrix semigroups, rhotrix rings, rhotrix Boolean algebra, rhotrix topological spaces, rhotrix metric spaces, rhotrix graphical trees called rhomtrees were developed. Furthermore, rhotrix finite fields, rhotrix exponent rule and their applications to special series, polynomial equations and polynomial rings over rhotrices were developed.

On the other hand, the contributory authors working on non-commutative rhotrix theory focus their researches majorly on extending the properties of matrices to rhotrices. Their inspirations came from the works of Sani   in his papers on conversion of rhotrix to a special matrix termed as coupled matrix. These articles made several authors to study analogous properties of matrices to rhotrices.

Now, it is also pertinent for us to mention here that authors use the same symbol “” to denote both heart based rhotrix multiplication and row-column based rhotrix multiplication in their research papers. That could confuse readers as per which of the multiplication method was intended, particularly, when an algebraic structure is denoted as a pair. So to ensure clarity, it would be better for interested authors to use the symbol “” to denote heart based rhotrix multiplication and the symbol “” to denote row-column based rhotrix multiplication in the future works.

In over all, we can say that from 2003 to 2013, the class of non-commutative rhotrix theory has more than 9% of articles in the literature of rhotrix theory than the class of commutative rhotrix theory.

4. Conclusion

In conclusion, we have presented a survey of articles on rhotrix theory starting from the year 2003 when the concept was initiated up to 2013. We have also classified the articles on rhotrix theory into two classes as commutative rhotrix theory and non-commutative rhotrix theory. It was shown in our analysis that the class of non- commutative rhotrix theory possessed 54.46% of articles in the literature of rhotrix theory while the class of commutative rhotrix theory possessed 45.24% of the articles.

Acknowledgements

We wish to thank the unknown reviewers for their helpful suggestions. We also wish to thank Ahmadu Bello University, Zaria, Nigeria for funding this relatively new area of research.

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