1. Introduction
By a rhotrix A of dimension three, we mean a rhomboidal array defined as
![](https://www.scirp.org/html/2-2230010\d8858f9d-b3be-4386-979b-2be3ae952c09.jpg)
where,
. The entry
in rhotrix
is called the heart of
and it is often denoted by
. The concept of rhotrix was introduced by [1] as an extension of matrix-tertions and matrix noitrets suggested by [2]. Since the introduction of rhotrix in [1], many researchers have shown interest on development of concepts for Rhotrix theory that are analogous to concepts in Matrix theory (see [3-9]). Sani [7] proposed an alternative method of rhotrix multiplication, by extending the concept of row-column multiplication of two dimensional matrices to three dimensional rhotrices, recorded as follows:
where,
and
belong to set of all three dimensional rhotrices,
.
The definition of rhotrix was later generalized by [6] to include any finite dimension
Thus; by a rhotrix A of dimension
we mean a rhomboidal array of cardinality
. Implying a rhotrix R of dimension n can be written as
![](https://www.scirp.org/html/2-2230010\bf1a1e0e-42fb-4672-99d6-6ed6f80dc201.jpg)
The element
and
are called the major and minor entries of R respectively. A generalization of row-column multiplication method for n-dimensional rhotrices was given by [8]. That is, given any n-dimensional rhotrices
and
, the multiplication of
and
is as follows:
![](https://www.scirp.org/html/2-2230010\dc090d0d-fc7a-434f-a556-4bc678cdf52b.jpg)
The method of converting a rhotrix to a special matrix called “coupled matrix” was suggested by [9]. This idea was used to solve systems of
and
matrix problems simultaneously. The concept of vectors and rhotrix vector spaces and their properties were introduced by [3] and [4] respectively. To the best of our knowledge, the concept of rank and linear transformation of rhotrix has not been studied. In this paper, we consider the rank of a rhotrix and characterize its properties. We also extend the idea to suggest the necessary and sufficient condition for representing rhotrix linear transformation.
2. Preliminaries
The following definitions will help in our discussion of a useful result in this section and other subsequent ones.
2.1. Definition
Let
be an n-dimensional rhotrix. Then,
is the
-entries called the major entries of ![](https://www.scirp.org/html/2-2230010\62ddd763-e962-40db-bf92-20f92e3512f7.jpg)
and
is the
-entries called the minor entries of
.
2.2. Definition 2.2 [7]
A rhotrix
of n-dimension is a coupled of two matrices
and
consisting of its major and minor matrices respectively. Therefore,
and
are the major and minor matrices of
.
2.3. Definition
Let
be an n-dimensional rhotrix. Then, rows and columns of
(
) will be called the major (minor) rows and columns of
respectively.
2.4. Definition
For any odd integer n, an
matrix
is called a filled coupled matrix if
for all
whose sum
is odd. We shall refer to these entries as the null entries of the filled coupled matrix.
2.5. Theorem
There is one-one correspondence between the set of all n-dimensional rhotrices over
and the set of all
filled coupled matrices over
.
3. Rank of a Rhotrix
Let
, the entries
and
in the main diagonal of the major and minor matrices of
respectively, formed the main diagonal of R. If all the entries to the left (right) of the main diagonal in
are zeros,
is called a right (left) triangular rhotrix. The following lemma follows trivially.
3.1. Lemma
Let
is a left (right) triangular rhotrix if and only if
and
are lower (upper) triangular matrices.
wang#title3_4:spProof
This follows when the rhotrix
is being rotated through 45˚ in anticlockwise direction.
In the light of this lemma, any n-dimensional rhotrix
can be reduce to a right triangular rhotrix by reducing its major and minor matrix to echelon form using elementary row operations. Recall that, the rank of a matrix
denoted by
is the number of non-zero row(s) in its reduced row echelon form. If
, we define rank of
denoted by
as:
. (3)
It follows from equation (3) that many properties of rank of matrix can be extended to the rank of rhotrix. In particular, we have the following:
3.2. Theorem
Let
and
, be any two n-dimensional rhotrices, where
Then 1)
;
2)
;
3)
;
4)
.
wang#title3_4:spProof
The first two statements follow directly from the definition. To prove the third statement, we apply the corresponding inequality for matrices, that is,
, where
is
and
is
. Thus,
![](https://www.scirp.org/html/2-2230010\80bbce1b-cdb5-410b-b645-c0bc86498e01.jpg)
For the last statement, consider
![](https://www.scirp.org/html/2-2230010\9f5829c0-45ab-4a8a-81bd-90244c469bd1.jpg)
3.3. Example
Let
.
Then, the filled coupled matrix of
is given by
.
Now reducing
to reduce row echelon form
, we obtain
which is a coupled of
and
matrices, i.e.
and
respectively.
Notice that,
![](https://www.scirp.org/html/2-2230010\70f3a48c-3840-4332-a284-ce39d9a92e65.jpg)
Hence,
.
4. Rhotrix Linear Transformation
One of the most important concepts in linear algebra is the concept of representation of linear mappings as matrices. If
and
are vector spaces of dimension
and
respectively, then any linear mapping
from
to
can be represented by a matrix. The matrix representation of
is called the matrix of
denoted by
. Recall that, if
is a field, then any vector space
of finite dimension
over
is isomorphic to
. Therefore, any
matrix over
can be considered as a linear operator on the vector space
in the fixed standard basis. Following this ideas, we study in this section, a rhotrix as a linear operator on the vector space
. Since the dimension of a rhotrix is always odd, it follow that, in representing a linear map
on a vector space
by a rhotrix, the dimension of
is necessarily odd. Therefore, throughout what follows, we shall consider only odd dimensional vector spaces. For any
and
be an arbitrary field, we find the coupled
of ![](https://www.scirp.org/html/2-2230010\c6894dd1-ace0-4b40-8596-8fa0e0c5ab5c.jpg)
and
by
![](https://www.scirp.org/html/2-2230010\45afa261-8880-488e-bfbf-395d0b75d7a4.jpg)
It is clear that
coincides with
and so, if
, any n-dimensional vector spaces ![](https://www.scirp.org/html/2-2230010\b11e5735-ae8f-4e0a-917d-f00ef0a24277.jpg)
and ![](https://www.scirp.org/html/2-2230010\5fa16dc5-ffdc-42be-aea5-672ccebec1d5.jpg)
is of dimensions
and
respectively. Less obviously, it can be seen that not every linear map
of
can be represented by a rhotrix in the standard basis. For instance, the map
![](https://www.scirp.org/html/2-2230010\e0c2b7ad-94a2-426f-9860-2e581c011931.jpg)
defined by
![](https://www.scirp.org/html/2-2230010\36aebb99-76ba-4ba3-8962-2a7b30b251d9.jpg)
is a linear mapping on
which cannot be represented by a rhotrix in the standard basis. The following theorem characterizes when a linear map
on
can be represented by a rhotrix.
4.1. Theorem
Let
and
be a field. Then, a linear map
can be represented by a rhotrix with respect to the standard basis if and only if
is defined as
![](https://www.scirp.org/html/2-2230010\d8979cf2-22cb-47f9-9ce9-ca5d02af12c8.jpg)
where
and
are any linear map on
and
respectively.
Proof:
Suppose
is defined by
![](https://www.scirp.org/html/2-2230010\72278c8d-974c-4ea7-aa92-dd539cff2397.jpg)
where,
and
are any linear map on
and
respectively, and consider the standard basis
. Note that, for
and
. Since
are linear maps,
. Thus,
(5)
Let
for
and ![](https://www.scirp.org/html/2-2230010\87ecc125-7572-45a2-bf43-b9025a3285e1.jpg)
for
. Then from (5), we have the matrix of
is
. (6)
This is a filled coupled matrix from which we obtain the rhotrix representation of
as
.
Conversely:
Suppose
has a rhotrix representation
in the standard basis. Then, the corresponding matrix representation of
is the filled coupled given in (6) above. Thus, we obtain the system
(7)
From this system, it follows that for each
we have the linear transformation
defined by
![](https://www.scirp.org/html/2-2230010\6f6b228b-0845-461f-b523-867fc6736732.jpg)
where,
and
are any linear map on
with
for
and
for
.
4.2. Example
Consider the linear mappings
define by
To find the rhotrix of
relative to the standard basis. We proceed by finding the matrices of
. Thus,
![](https://www.scirp.org/html/2-2230010\d3a95f4d-f35e-410f-acdc-65eaf67c6485.jpg)
Therefore, by definition of matrix of
with respect to the standard basis, we have
![](https://www.scirp.org/html/2-2230010\dbff1d0a-8b37-4428-be7b-9dfbf2e56cf1.jpg)
which is a filled coupled matrix from which we obtain the rhotrix of
in
,
.
Now starting with the rhotrix
the filled coupled matrix of
is
.
And so, defining ![](https://www.scirp.org/html/2-2230010\912a3b2b-a896-4af7-ac0b-6fe084f3752e.jpg)
![](https://www.scirp.org/html/2-2230010\4490a14b-589d-4930-80cb-cfa3e11d90b1.jpg)
Thus, if
Therefore,
![](https://www.scirp.org/html/2-2230010\f2284354-c523-4c31-825c-faeb26fd616d.jpg)
5. Conclusion
We have considered the rank of a rhotrix and characterize its properties as an extension of ideas to the rhotrix theory rhomboidal arrays. Furthermore, a necessary and sufficient condition under which a linear map can be represented over rhotrix had been presented.
6. Acknowledgements
The Authors wish to thank Ahmadu Bello University, Zaria, Nigeria for financial support towards publication of this article.