This paper considers rank of a rhotrix and characterizes its properties, as an extension of ideas to the rhotrix theory rhomboidal arrays, introduced in 2003 as a new paradigm of matrix theory of rectangular arrays. Furthermore, we present the necessary and sufficient condition under which a linear map can be represented over rhotrix.
Rhotrix; Rank; Rhotrix Rank; Linear Transformation; Rhotrix Linear Transformation1. Introduction
By a rhotrix A of dimension three, we mean a rhomboidal array defined as
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
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:
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.
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:
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,
For the last statement, consider
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,
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 coupledof
and
by
It is clear that coincides with and so, if, any n-dimensional vector spaces
and 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
defined by
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
where and are any linear map on and respectively.
Proof:
Suppose is defined by
where, and are any linear map on and respectively, and consider the standard basis
. Note that, for
and. Since are linear maps,. Thus,
Let for
and
for. Then from (5), we have the matrix of is
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
From this system, it follows that for each we have the linear transformation defined by
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,
Therefore, by definition of matrix of with respect to the standard basis, we have
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
Thus, if
Therefore,
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.
REFERENCESReferencesA. O. Ajibade, “The Concept of Rhotrix in Mathematical Enrichment,” International Journal of Mathematical Education in Science and Technology, Vol. 34, No. 2, 2003, pp. 175-179. doi:10.1080/0020739021000053828K. T. Atanassov and A. G. Shannon, “Matrix-Tertions and Matrix-Noitrets: Exercise for Mathematical Enrichment,” International Journal of Mathematical Education in Science and Technology, Vol. 29, No. 6, 1998, pp. 898- 903.A. Aminu, “Rhotrix Vector Spaces,” International Journal of Mathematical Education in Science and Technology, Vol. 41, No. 4, 2010, pp. 531-573.
doi:10.1080/00207390903398408A. Aminu, “The Equation Rnx = b over Rhotrices,” International Journal of Mathematical Education in Science and Technology, Vol. 41, No. 1, 2010, pp. 98-105.
doi:10.1080/00207390903189187A. Mohammed, “Enrichment Exercises through Extension to Rhotrices,” International Journal of Mathematical Education in Science and Technology, Vol. 38, No. 1, 2007, pp. 131-136. doi:10.1080/00207390600838490A. Mohammed, “Theoretical Development and Applications of Rhotrices,” Ph.D. Thesis, Ahmadu Bello University, Zaria, 2011.B. Sani, “An Alternative Method for Multiplication of Rhotrices,” International Journal of Mathematical Education in Science and Technology, Vol. 35, No. 5, 2004, pp. 777-781. doi:10.1080/00207390410001716577B. Sani, “The Row-Column Multiplication of Higher Dimensional Rhotrices,” International Journal of Mathematical Education in Science and Technology, Vol. 38, No. 5, 2007, pp. 657-662.B. Sani, “Conversion of a Rhotrix to a ‘Coupled Matrix’,” International Journal of Mathematical Education in Science and Technology, Vol. 39, No. 2, 2008, pp. 244-249.
doi:10.1080/00207390701500197
. The entry 
in rhotrix 
is called the heart of 
and it is often denoted by
. The concept of rhotrix was introduced by [
where, 
and 
belong to set of all three dimensional rhotrices,
.
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
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 [
and
, the multiplication of 
and 
is as follows:
and 
matrix problems simultaneously. The concept of vectors and rhotrix vector spaces and their properties were introduced by [
be an n-dimensional rhotrix. Then,
is the 
-entries called the major entries of 
is the 
-entries called the minor entries of
.
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
.
be an n-dimensional rhotrix. Then, rows and columns of 
(
) will be called the major (minor) rows and columns of 
respectively.
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.
and the set of all 
filled coupled matrices over
.
, 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.
is a left (right) triangular rhotrix if and only if 
and 
are lower (upper) triangular matrices.
is being rotated through 45˚ in anticlockwise direction.
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:
and
, be any two n-dimensional rhotrices, where 
Then 1)
;
;
;
.
, where 
is 
and 
is
. Thus,
.
is given by
.
to reduce row echelon form 
, we obtain
which is a coupled of 
and 
matrices, i.e. 
and 
respectively.
.
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 
and
by
coincides with 
and so, if
, any n-dimensional vector spaces 
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
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.
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
and 
are any linear map on 
and 
respectively.
is defined by
and 
are any linear map on 
and 
respectively, and consider the standard basis
. Note that, for
and
. Since 
are linear maps,
. Thus,
for
and 
. Then from (5), we have the matrix of 
is
as
.
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
we have the linear transformation 
defined by
and 
are any linear map on 
with 
for 
and 
for
.
define by 
To find the rhotrix of 
relative to the standard basis. We proceed by finding the matrices of
. Thus,
with respect to the standard basis, we have
in
,
.
the filled coupled matrix of 
is
.
