**Advances in Linear Algebra & Matrix Theory** Vol.2 No.4(2012), Article ID:25471,4 pages DOI:10.4236/alamt.2012.24007

Rhotrix Linear Transformation

Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria

Email: abdulmaths@yahoo.com

Received September 20, 2012; revised October 26, 2012; accepted November 6, 2012

**Keywords:** Rhotrix; rank; rhotrix rank; linear transformation; rhotrix linear transformation

ABSTRACT

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.

1. 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.

2.1. Definition

Let be an n-dimensional rhotrix. Then,

is the -entries called the major entries of

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,

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,

(5)

Let for

and

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

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.

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