^{1}

^{*}

^{1}

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.

In the year 2003, a relatively new paradigm of science, now known as rhotrix theory was initiated by Ajibade [

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 [

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.

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.

In [

where

S/no. | Title |
---|---|

1 | A note on the rhotrix system of equations |

2 | A note on rhotrix exponent rule and its applications to special series and polynomial equation defined over rhotrices |

3 | A remark on the classifications of rhotrices as abstract structures |

4 | Algebraic properties of singleton, coiled and modulo rhotrices |

5 | Certain field of fractions |

6 | Certain quadratic extensions |

7 | Enrichment exercises through extension to rhotrices |

8 | Generalization and algorithmatization of heart based method for multiplication of rhotrices |

9 | Note on certain field of fractions |

10 | Note on rhotrices and the construction of finite fields |

11 | On construction of rhomtrees as graphical representation of rhotrices |

12 | On the structure of rhotrix |

13 | On the linear system over rhotrices |

14 | Rhotrices and the construction of finite fields |

15 | Rhotrix polynomials and polynomial rhotrices |

16 | Rhotrix sets and rhotrix spaces category |

17 | Rhotrix topological spaces |

18 | The concept of rhotrix in mathematical enrichment |

19 | The concept of heart oriented rhotrix multiplication |

be the set of all real rhotrices of size

on division of

any two rhotrices in

(4)

where

The extended rhotrix multiplication (4) was named in [

A remark on classifications of rhotrices as abstract structures was proposed by [

The generalization of Ajibade’s heart based method for rhotrix multiplication in [

The concept of tree in graph theory was extended to rhotrix theory by [

rhomtrees of order

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 [

Sani [

S/no. | Titles |
---|---|

1 | A determinant method for solving rhotrix system of eqn. |

2 | A note on relationship between invertible rhotrices and associated invertible matrices |

3 | Adjacent rhotrix of a complete, simple and undirected graph |

4 | Adjoint of a rhotrix and its basic properties |

5 | Algorithm design for row-column multiplication of n-dimensional rhotrices |

6 | An alternative method for multiplication of rhotrices |

7 | An example of linear mappings: extension to rhotrices |

8 | Cayley-Hamilton theorem in rhotrix |

9 | Conversion of a rhotrix to a coupled matrix |

10 | Hilbert matrix and its relationship with a special rhotrix |

11 | On inner product space and bilinear forms over rhotrices |

12 | On involutory and Pascal rhotrices |

13 | On the construction of involutory rhotrices. |

14 | Parallel multiplication of rhotrices using systolic array architecture |

15 | Rhotrix multiplication on two-dimensional process grid topologies |

16 | Rhotrices and elementary row operations |

17 | Rhotrix linear transformation |

18 | Rhotrix vector spaces |

19 | Row-wise representation of arbitrary rhotrix |

20 | Solution of two coupled matrices |

21 | The Cayley-Hamilton theorem for rhotrices |

22 | The equation |

23 | The row-column multiplication of high dimensional rhotrices |

section in [

This multiplication was later generalized by [

where

This (6) was presented in [

In [

A method of converting rhotrix to a special form of matrix called “coupled matrix” was given in [

where

This is a rhotrix

Two coupled matrices

If a coupled matrix

the matrices

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

Sani [

A one-sided system of the form

The rhotrix addition and scalar multiplication defined in [

Following this, [

In [

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 [

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 [

S/no. | Category | Papers | Percentage (%) |
---|---|---|---|

1 | Class of commutative rhotrix theory | 19 | 45.24 |

2 | Class of non-commutative rhotrix theory | 23 | 54.46 |

Total | 42 | 100 |

theorems on involution in the context of rhotrices. Also, the description of Pascal rhotrices and their related properties was also considered. In [

In this section, we present two tables for the analysis of articles in the literature review of rhotrix theory. In

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 [

Now, it is also pertinent for us to mention here that authors use the same symbol “

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.

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.

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.