# On the Representation of Neutrosophic Matrices by Neutrosophic Linear Transformations

Authors: Mohammad Abobala
Journal: Journal of Mathematics
 (2021)

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## Abstract

The objective of this paper is to study the representation of neutrosophic matrices defined over a neutrosophic field by neutrosophic linear transformations between neutrosophic vector spaces, where it proves that every neutrosophic matrix can be represented uniquely by a neutrosophic linear transformation. Also, this work proves that every neutrosophic linear transformation must be an AH-linear transformation; i.e., it can be represented by classical linear transformations.


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## Body

## 1. Introduction

Neutrosophy is a new branch of philosophy founded by Smarandache [1, 2] to deal with uncertainty in real-life problems.Neutrosophic concepts found their way in many other fields, such as classification [3, 4], number theory [5, 6], algebraic equations [7, 8], Boolean algebra [9] and optimization [10].Neutrosophic algebra began with Smarandache and Kandasamy in [11], where they defined neutrosophic rings and fields for the first time. Lately, neutrosophic fields [12] were used in the study of neutrosophic vector spaces [13–16].Neutrosophic matrices were defined to deal with indeterminacy problems, and many applications and theorems can be found in [17–19].If V is a vector space over the field F, thenVI=x+yI;x,y∈V is the corresponding strong neutrosophic vector space over the neutrosophic field F(I).In [4, 20–24], Abobala et al. proposed the concept of AH substructures in groups, rings, spaces, and modules as a neutrosophic structures with two classical parts; for example, in the strong neutrosophic vector space V(I), an AH subspace is the set WI=T+SI, where TandS are two classical subspaces of V. In a similar way, an AH linear transformation is a function f between two neutrosophic vector spaces VIandWI with two classical parts f=g+hI, where gandh are classical linear transformations between V and W.It is known that classical matrices can be represented by linear transformations; from this point of view, we will study this problem in single valued neutrosophic systems.In this work, we study neutrosophic matrices as linear neutrosophic functions. In particular, we prove that every linear transformation between two neutrosophic vector spaces must have an AH structure.


## 2. Preliminaries

Definition 1 (see [16]).
Let (V, +, .) be a vector space over the field K, then (V(I), +, .) is called a weak neutrosophic vector space over the field K, and it is called a strong neutrosophic vector space if it is a vector space over the neutrosophic field K(I).
A neutrosophic fieldK(I) is a triple (K(I), +, .), where K is a classical field. A neutrosophic field is not a field by classical meaning, but it is a ring.
Elements ofV(I) have the following form: x+yI;x,y∈V; i.e., V(I) can be written as VI=V+VI.Definition 2 (see [16]).
LetV(I) be a strong neutrosophic vector space over the neutrosophic field K(I) and W(I) be a nonempty set of V(I), then W(I) is called a strong neutrosophic subspace if W(I) itself is a strong neutrosophic vector space.Definition 3 (see [16]).
Letv1,v2.vs∈VI,andx∈VI we say that x is a linear combination of vi;i=1,..,s if(1)x=a1v1+…+asvssuchai∈KI.
The setvi;i=1,..,s is called linearly independent if a1v1+⋯+asvs=0 implies ai=0 for all i.Definition 4 (see [18]).
LetMm×n=aij:aij∈KI where KI is a neutrosophic field. We call it the neutrosophic matrix.


## 3. Main Discussion

Theorem 1.
LetV,W be two vector spaces over the field F with dimV=n,dimW=m and VI,WI be the corresponding neutrosophic vector spaces over the corresponding neutrosophic field FI. Let g,h:V⟶W be two linear transformations, then there exists a neutrosophic linear transformation f=g+hI:VI⟶WI, where f is defined as follows:(2)fx+yI=gx+g+hx+y−gxI.Proof.
We definef=g+hI:VI⟶WI, where(3)fx+yI=gx+g+hx+y−gxI,in which f is a linear transformation, that is, because for every m=x+yI,n=z+tI∈VI, we have(4)fm+n=fx+z+Iy+t=gx+z+Ig+hx+y+z+t−gx+z=gx+g+hx+y−gxI+gz+g+hz+t−gzI=fm+fn.
On the contrary, consider an arbitrary neutrosophic numbera+bI∈FI, then(5)fa+bIm=fa+bIx+yI=fax+Iay+bx+by=fax+Ia+bx+y−ax=gax+Ig+ha+bx+y−gax=agx+Ia+bg+hx+y−agx=a+bIgx+Ig+hx+y−gx=a+bIfm.
Thus,f is a neutrosophic linear transformation.Definition 5.
The neutrosophic linear transformationf defined in Theorem 1 is called a full AH-linear transformation.Definition 6.
Letf=g+hI:VI⟶WI be a full AH-linear transformation and M=A+BI be an n×m neutrosophic matrix over FI, and we call M the neutrosophic matrix of f if and only if fx+yI=Mx+yI for every x+yI∈VI.Theorem 2.
Letf=g+hI:VI⟶WI be any full AH-linear transformation, then M=A+BI is the corresponding neutrosophic matrix if and only if A is the matrix of g and B is the matrix of h.Proof.
We assume that A is the matrix ofg and B is the matrix of h; hence, Ax=gx,By=hy,A+Bx+y=g+hx+y. We have(6)M.x+yI=A+BIx+yI=Ax+IAy+Bx+By=Ax+IA+Bx+y−Ax=gx+Ig+hx+y−gx=fx+yI.
Thus,M is the neutrosophic matrix of f.
Conversely, suppose thatM is the neutrosophic matrix of f, and we shall prove that A is the matrix of g and B is the matrix of h.
According to the assumption, we haveMx+yI=fx+yI; hence,(7)Ax+IA+Bx+y−Ax=gx+Ig+hx+y−gx.
This implies thatAx=gx,A+Bx+y=g+hx+y so that Bx+y=hx+y. By considering the arbitrariness of x and y, we get that A is the matrix of g and B is the matrix of h.Example 1.
(a)
LetVI=R2I=a,b+c,dI=a+cI,b+dI;a,b,c,d∈R, consider the following neutrosophic matrix M=1+II−I2−I. The corresponding neutrosophic linear transformation is defined as follows:(8)fx+yI=M.a+cIb+dI=a+Ic+a+c+b+d,−aI−cI+2b+2dI−bI−dI=a+Ia+2c+b+d,2b+I−a−c−b+d=a,2b+Ia+2c+b+d,−a−c−b+d.(b)
f=g+hI;gx,y=x,2y,hx,y=x+y,−x−y, where g,h:V⟶V.Theorem 3.
LetV,W be two vector spaces over the field F, with dimV=n,dimW=m, and let M=A+BI be any n×m neutrosophic matrix over FI. Then, M can be represented by a unique full AH-linear transformation f=g+hI, where A is the matrix of g and B is the matrix of h.Proof.
According to Theorem2, the neutrosophic matrix M=A+BI can be represented by a neutrosophic full AH-linear transformation f=g+hI, where A is the matrix of g and B is the matrix of h. For the uniqueness condition, we suppose that F=G+HI is another linear AH-transformation with the property.
Mx+yI=Fx+yI. We have(9)M.x+yI=Fx+yI=fx+yI,for allx+yI∈VI.
Thus,F=f and f is unique.
The following theorem shows an algorithm to find a basis for the neutrosophic vector spaceV(I) from any basis of the corresponding classical vector space V.Theorem 4.
LetV(I) be any neutrosophic vector space over the neutrosophic field F(I) and V be its corresponding classical vector space over the field F. Let S={v1,v2,…,vn} be a basis of V over F, then L=lij=vi+vj−viI;1≤i,j≤n is a basis of V(I) over F(I).Proof.
First of all, we must prove thatL generates V(I) over F(I). Let x+yI be any element of V (I), where x,y∈V, and we have(10)x=∑i=1naivi,x+y=∑j=1nbjvj,We putrij=ai+bj−aiI∈FI.
Now, we compute∑i,j=1nrijlij.(11)∑i,j=1nrijlij=∑i,j=1nai+bj−aiIvi+vj−viI=∑i,j=1naivi+Ibjvj−aivi=∑i=1naivi+I∑j=1nbjvj−∑i=1naivi=x+Ix+y−x=x+yI.
Thus,L generates V(I) over F(I).
Now, we prove thatL is linearly independent. For this purpose, we assume that ∑i,j=1nai+bjIlij=0; thus, we get(12)∑i,j=1naivi+Iai+bjvj−aivi=0,hence,∑i=1naivi=∑i,jnai+bjvj=0,thus,ai=ai+bj=0,so that,bj=0.
This implies thatL is linearly independent, and then it is a basis.Example 2.
It is well known that {x = (1, 0), y = (0, 1)} is a basis of V = R2. The corresponding basis of V(I) = R2I is(13)x,y,x+y−xI,y+x−yI=1,0,0,1,1,0+−1,1I,0,1+1,−1I.
The following theorem shows that every linear transformation betweenVI and WI must be a full AH-linear transformation.Theorem 5.
LetV,W be two vector spaces over the field F, with dimV=n,dimW=m and let VI,WI be the corresponding neutrosophic vector spaces over FI. Let f:VI⟶WI be any linear transformation, then f is a full AH-linear transformation.Proof.
Letf:VI⟶WI be any linear transformation, and we must prove that there exists two classical linear transformations g,q:V⟶W, where f=g+qI.
Suppose thatS={v1,v2,…,vn} is a basis of V, then L=lij=vi+vj−viI;1≤i,j≤n is a basis of V (I). It is known that fL=fvi+vj−viI=wi+wj−wiI;wi,wj∈W is a basis of W (I) and that is because the direct image of a basis by any linear transformation is a gain a basis.
Defineg:V⟶W;gvi=wi,h:V⟶W;hvj=wj. It is clear that fvi+vj−viI=gvi+Ihvj−gvi. This means that f=g+qI=g+h−gI. Now, we must prove that g,q=h−g are classical linear transformations.
Letx,y be any two elements of V, we have x=x+0I,y=y+0I∈VI. We have(14)fx+y=fx+0I+y+0I=gx+y=gx+gy.
For anym∈F,we havem=m+0I∈FI,andfm+0Ix+0I=fmx+0I=gmx=mgx, and thus, g is a linear transformation.
On the other hand, we have(15)xI,yI∈VI,andfxI+yI=fx+yI=f0+x+yI=g0+Ig+qx+y−g0=Ihx+y=hxI+hyI,thushx+y=hx+hy,fm+0I0+xI=f0+mxI=g0+Ig+qmx−g0=Ihmx=mhxI,so thathmx=mhx.
This implies thatg,h are two classical linear transformations; thus, g,q are linear transformations, which implies that f=g+qI is a full AH-linear transformation.Remark 1.
From Theorem5 and Theorem 3, we get the following interesting result: every neutrosophic linear transformation f:VI⟶WI can be represented by a unique neutrosophic matrix M=A+BI.


### 3.1. Further Applications

According to this work, we can use linear functions to study any problem that needs neutrosophic matrices. From this point of view, single-valued neutrosophic matrices used in [19] can be turned into algebraic linear functions.


## 3.1. Further Applications

According to this work, we can use linear functions to study any problem that needs neutrosophic matrices. From this point of view, single-valued neutrosophic matrices used in [19] can be turned into algebraic linear functions.


## 4. Conclusion

In this paper, we have proved that every neutrosophic matrix can be represented uniquely by a neutrosophic linear vector space transformation. Also, we have showed that the linear property of any neutrosophic vector space function implies the AH-structure of this function.This work opens a wide door to use neutrosophic vector spaces and matrices in classical representation theory of groups since it is well known that classical groups are represented by linear transformations from a vector space to itself. According to our results, we can find an important application of neutrosophic algebraic theory in the classical representation theory of groups. This application can be summarized by the following open question.


## 5. Open Problem

Determine the algebraic structure of all groups which can be represented by neutrosophic linear transformations from a neutrosophic vector spaceV(I) to itself.



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*Source: 5591576-2021-02-25.xml*