_{1}

In this article, starting from geometrical considerations, he was born with the idea of 3D matrices, which have developed in this article. A problem here was the definition of multiplication, which we have given in analogy with the usual 2D matrices. The goal here is 3D matrices to be a generalization of 2D matrices. Work initially we started with 3×3×3 matrix, and then we extended to
*m*×
*n*×
*p *matrices. In this article, we give the meaning of 3D matrices. We also defined two actions in this set. As a result, in this article, we have reached to present 3-dimensional unitary ring matrices with elements from a field F.

Based on the meaning of the addition and the multiplication of 2D matrices [

Imagining a parallelepiped, with born idea of 3D matrices, which are define as follows

Definition 2.1 3-dimensional 3 × 3 × 3 matrice will call, a matrix which has: three horizontal layers (analogous to three rows), three vertical page (analogue with three columns in the usual matrices) and three vertical layers two of which are hidden.

The set of these matrices the write how:

M 3 × 3 × 3 ( F ) = { ( a i j k ) | a i j k ∈ F and i = 1 , 2 , 3 ; j = 1 , 2 , 3 ; k = 1 , 2 , 3 }

The appearance of these matrices will be as in

Definition 2.2 The addition of two matrices A 3 × 3 × 3 , B 3 × 3 × 3 ∈ M 3 × 3 × 3 ( F ) we will call the matrix:

C 3 × 3 × 3 = { ( c i j k ) | c i j k = a i j k + b i j k , ∀ i , j , k ∈ { 1 , 2 , 3 } }

The appearance of the addition of 3 × 3 × 3, 3D matrices, will be as in

A 3 × 3 × 3 = { ( a i j k ) | a i j k ∈ F for i = 1 , 2 , 3 ; j = 1 , 2 , 3 ; k = 1 , 2 , 3 } B 3 × 3 × 3 = { ( b i j k ) | b i j k ∈ F for i = 1 , 2 , 3 ; j = 1 , 2 , 3 ; k = 1 , 2 , 3 }

Definition 2.3 Zero matrix 3 × 3, 3D we will called the matrix that has all its elements zero.

O 3 × 3 × 3 = { ( 0 F ) i j k | i = 1 , 2 , 3 ; j = 1 , 2 , 3 ; k = 1 , 2 , 3 }

Definition 2.4. The opposite matric of anmatrice

A 3 × 3 × 3 = { ( a i j k ) | i = 1 , 2 , 3 ; j = 1 , 2 , 3 ; k = 1 , 2 , 3 }

will, called matrix

− A 3 × 3 × 3 = { ( − a i j k ) | i = 1 , 2 , 3 ; j = 1 , 2 , 3 ; k = 1 , 2 , 3 }

(where − a i j k is a opposite element of element a i j k ∈ F , so a i j k + ( − a i j k ) = 0 F and ( F , + , ⋅ ) is field [

A 3 × 3 × 3 + ( − A 3 × 3 × 3 ) = { ( a i j k + ( − a i j k ) ) | i = 1 , 2 , 3 ; j = 1 , 2 , 3 ; k = 1 , 2 , 3 } = { ( 0 ) i j k | i = 1 , 2 , 3 ; j = 1 , 2 , 3 ; k = 1 , 2 , 3 } = O 3 × 3 × 3

Theorem 2.1 ( M 3 × 3 × 3 ( F ) , + ) is a beliangrup.

Proof: Truly from the definition 2.2, of addition the 3-Dmatrices, we see that addition is the sustainable in M 3 × 3 × 3 ( F ) , because

a i j k ∈ F , b i j k ∈ F ⇒ c i j k = a i j k + b i j k ∈ F , ∀ i , j , k ∈ { 1 , 2 , 3 }

1) Associative property,

∀ A = ( a i j k ) , B = ( b i j k ) , C = ( c i j k ) ∈ M 3 × 3 × 3 ( F ) ⇒ ( A + B ) + C = A + ( B + C )

truly

( A + B ) + C = [ ( a i j k ) + ( b i j k ) ] + ( c i j k ) = ( a i j k + b i j k ) + ( c i j k ) = ( ( a i j k + b i j k ) + c i j k ) = ( a i j k + b i j k + c i j k ) = ( a i j k + ( b i j k + c i j k ) ) = ( a i j k ) + ( b i j k + c i j k ) = A + ( B + C )

2) ∀ A = ( a i j k ) ∈ M 3 × 3 × 3 ( F ) , ∃ O = ( 0 i j k ) / A + O = O + A = A .

truly, ∀ A = ( a i j k ) ∈ M 3 × 3 × 3 ( F ) , ∃ O = ( 0 i j k ) / A + O = O + A = A .

A + O = { ( a i j k ) + ( 0 ) i j k | i = 1 , 2 , 3 ; j = 1 , 2 , 3 ; k = 1 , 2 , 3 } = { ( a i j k + 0 ) i j k | i = 1 , 2 , 3 ; j = 1 , 2 , 3 ; k = 1 , 2 , 3 } = { ( a i j k ) | i = 1 , 2 , 3 ; j = 1 , 2 , 3 ; k = 1 , 2 , 3 } = A

3) ∀ A = ( a i j k ) ∈ M 3 × 3 × 3 ( F ) , ∃ − A = ( − a i j k ) ∈ M 3 × 3 × 3 ( F ) / A + ( − A ) = O .

truly, from Definition 2.4, we have

A + ( − A ) = { ( a i j k + ( − a i j k ) ) | i = 1 , 2 , 3 ; j = 1 , 2 , 3 ; k = 1 , 2 , 3 } = { ( 0 ) i j k | i = 1 , 2 , 3 ; j = 1 , 2 , 3 ; k = 1 , 2 , 3 } = O

4) Addition is commutative.

∀ A = ( a i j k ) , B = ( b i j k ) ∈ M 3 × 3 × 3 ( F ) , A + B = B + A .

truly

A + B = ( a i j k ) + ( b i j k ) = ( a i j k + b i j k ) = ( ℝ , + ) isabelian ( b i j k + a i j k ) = ( b i j k ) + ( a i j k ) = B + A

Definition 3.1 3-dimensionalmxnxp matrix will call, a matrix which has: m- horizontal layers (analogous to m-rows), n-vertical page (analogue with n-col- umns in the usualmatrices) and p-vertical layers (p − 1 of which are hidden).

The set of these matrixes the write how:

M m × n × p ( F ) = { ( a i j k ) | a i j k ∈ F -fieldand i = 1 , m ¯ ; j = 1 , n ¯ ; k = 1 , p ¯ }

Definition 3.2 The addition of two matrices A , B ∈ M m × n × p ( F ) we will call the matrix:

C m × n × p = { ( c i j k ) | c i j k = a i j k + b i j k , ∀ i = 1 , m ¯ ; j = 1 , n ¯ ; k = 1 , p ¯ }

The appearance of the addition of mxnxp, 3D matrices will be as in

A m × n × p = { ( a i j k ) | i = 1 , m ¯ ; j = 1 , n ¯ ; k = 1 , p ¯ } B m × n × p = { ( b i j k ) | i = 1 , m ¯ ; j = 1 , n ¯ ; k = 1 , p ¯ }

Definition 3.3 3-D, Zero matrix m × n × p , we will called the matrix that has all its elements zero.

O = O m × n × p = { ( 0 ) i j k | i = 1 , m ¯ ; j = 1 , n ¯ ; k = 1 , p ¯ }

Definition 3.4 The opposite matric of anmatrice

A m × n × p = { ( a i j k ) | i = 1 , m ¯ ; j = 1 , n ¯ ; k = 1 , p ¯ } ∈ M m × n × p ( F )

will, called matrix

− A m × n × p = { ( − a i j k ) | i = 1 , m ¯ ; j = 1 , n ¯ ; k = 1 , p ¯ } ∈ M m × n × p ( F )

(where − a i j k is a opposite element of element a i j k ∈ F , so a i j k + ( − a i j k ) = 0 F and ( F , + , ⋅ ) is field), which satisfies the condition

A m × n × p + ( − A m × n × p ) = { ( a i j k + ( − a i j k ) ) | i = 1 , m ¯ ; j = 1 , n ¯ ; k = 1 , p ¯ } = { ( 0 ) i j k | i = 1 , m ¯ ; j = 1 , n ¯ ; k = 1 , p ¯ } = O

Theorem 3.1 ( M m × n × p ( F ) , + ) is abeliangrup.

Proof: Truly from the definition 3.2, of additions the 3-D matrices, we see that addition is the sustainable in M m × n × p ( F ) , because

a i j k ∈ F , b i j k ∈ F ⇒ c i j k = a i j k + b i j k ∈ F , ∀ i = 1 , m ¯ ; j = 1 , n ¯ ; k = 1 , p ¯

1) Associative property,

∀ A = ( a i j k ) , B = ( b i j k ) , C = ( c i j k ) ∈ M m × n × p ( F ) ⇒ ( A + B ) + C = A + ( B + C )

truly

( A + B ) + C = [ ( a i j k ) + ( b i j k ) ] + ( c i j k ) = ( a i j k + b i j k ) + ( c i j k ) = ( ( a i j k + b i j k ) + c i j k ) = ( a i j k + b i j k + c i j k ) = ( a i j k + ( b i j k + c i j k ) ) = ( a i j k ) + ( b i j k + c i j k ) = A + ( B + C )

2) ∀ A = ( a i j k ) ∈ M m × n × p ( F ) , ∃ O = ( 0 i j k ) ∈ M m × n × p ( F ) / A + O = O + A = A .

truly,

∀ A = ( a i j k ) ∈ M m × n × p ( F ) , ∃ O = ( 0 i j k ) ∈ M m × n × p ( F ) / A + O = O + A = A .

A + O = { ( a i j k ) + ( 0 ) i j k | i = 1 , m ¯ ; j = 1 , n ¯ ; k = 1 , p ¯ } = { ( a i j k + 0 F ) i j k | i = 1 , m ¯ ; j = 1 , n ¯ ; k = 1 , p ¯ } = { ( a i j k ) | i = 1 , m ¯ ; j = 1 , n ¯ ; k = 1 , p ¯ } = A

3) ∀ A = ( a i j k ) ∈ M m × n × p ( F ) , ∃ − A = ( − a i j k ) ∈ M m × n × p ( F ) / A + ( − A ) = O .

truly, from Definition 2.4, we have

A + ( − A ) = { ( a i j k + ( − a i j k ) ) | i = 1 , m ¯ ; j = 1 , n ¯ ; k = 1 , p ¯ } = { ( 0 F ) i j k | i = 1 , m ¯ ; j = 1 , n ¯ ; k = 1 , p ¯ } = O

4) Addition is commutative.

∀ A = ( a i j k ) , B = ( b i j k ) ∈ M m × n × p ( F ) / A + B = B + A .

truly

A + B = ( a i j k ) + ( b i j k ) = ( a i j k + b i j k ) = ( F , + ) isabelian ( b i j k + a i j k ) = ( b i j k ) + ( a i j k ) = B + A

Definition 4.1: The multiplication of two matrices A , B ∈ M 3 × 3 × 3 ( F ) we will call the matrix C = A ⊗ B ∈ M 3 × 3 × 3 ( F ) calculated as follows:

∀ | 3-verticallayer 2-verticallayer 1-verticallayer ( a 113 a 123 a 133 a 213 a 223 a 233 a 313 a 323 a 333 a 112 a 122 a 132 a 212 a 222 a 232 a 312 a 322 a 332 a 111 a 121 a 131 a 211 a 221 a 231 a 311 a 321 a 331 ) , 3-verticallayer 2-verticallayer 1-verticallayer ( b 113 b 123 b 133 b 213 b 223 b 233 b 313 b 323 b 333 b 112 b 122 b 132 b 212 b 222 b 232 b 312 b 322 b 332 b 111 b 121 b 131 b 211 b 221 b 231 b 311 b 321 b 331 ) ∈ M 3 × 3 × 3 ( F )

The appearance of the multiplication of 3 × 3 × 3, 3D matrices will be as in

C = ( 3-verticallayer ) ( 2-verticallayer ) ( 1-verticallayer ) ( c 113 c 123 c 133 c 213 c 223 c 233 c 313 c 323 c 333 c 112 c 122 c 132 c 212 c 222 c 232 c 312 c 322 c 332 c 111 c 121 c 131 c 211 c 221 c 231 c 311 c 321 c 331 ) = ( a 113 a 123 a 133 a 213 a 223 a 233 a 313 a 323 a 333 a 112 a 122 a 132 a 212 a 222 a 232 a 312 a 322 a 332 a 111 a 121 a 131 a 211 a 221 a 231 a 311 a 321 a 331 ) ⊗ ( b 113 b 123 b 133 b 213 b 223 b 233 b 313 b 323 b 333 b 112 b 122 b 132 b 212 b 222 b 232 b 312 b 322 b 332 b 111 b 121 b 131 b 211 b 221 b 231 b 311 b 321 b 331 )

where, the first vertical page is:

c 111 = a 111 ⋅ b 111 + a 121 ⋅ b 211 + a 131 ⋅ b 311 ; c 112 = a 112 ⋅ b 112 + a 122 ⋅ b 211 + a 132 ⋅ b 312 ; c 113 = a 113 ⋅ b 113 + a 123 ⋅ b 213 + a 133 ⋅ b 313 ;

c 211 = a 211 ⋅ b 111 + a 221 ⋅ b 211 + a 231 ⋅ b 311 ; c 212 = a 212 ⋅ b 112 + a 222 ⋅ b 211 + a 232 ⋅ b 312 ; c 213 = a 213 ⋅ b 113 + a 223 ⋅ b 213 + a 233 ⋅ b 313 ;

c 311 = a 311 ⋅ b 111 + a 321 ⋅ b 211 + a 331 ⋅ b 311 ; c 312 = a 312 ⋅ b 112 + a 322 ⋅ b 211 + a 332 ⋅ b 312 ; c 313 = a 313 ⋅ b 113 + a 323 ⋅ b 213 + a 333 ⋅ b 313 ;

the second vertical page is:

c 121 = a 111 ⋅ b 121 + a 121 ⋅ b 221 + a 131 ⋅ b 321 ; c 122 = a 112 ⋅ b 122 + a 122 ⋅ b 222 + a 132 ⋅ b 322 ; c 123 = a 113 ⋅ b 123 + a 123 ⋅ b 223 + a 133 ⋅ b 323 ;

c 221 = a 211 ⋅ b 121 + a 221 ⋅ b 221 + a 231 ⋅ b 321 ; c 222 = a 212 ⋅ b 122 + a 222 ⋅ b 222 + a 232 ⋅ b 322 ; c 223 = a 213 ⋅ b 123 + a 223 ⋅ b 223 + a 233 ⋅ b 323 ;

c 321 = a 311 ⋅ b 121 + a 321 ⋅ b 221 + a 331 ⋅ b 321 ; c 322 = a 312 ⋅ b 122 + a 322 ⋅ b 222 + a 332 ⋅ b 322 ; c 323 = a 313 ⋅ b 123 + a 323 ⋅ b 223 + a 333 ⋅ b 323 ;

and third vertical page is:

c 131 = a 111 ⋅ b 131 + a 121 ⋅ b 231 + a 131 ⋅ b 331 ; c 132 = a 112 ⋅ b 132 + a 122 ⋅ b 232 + a 132 ⋅ b 332 ; c 133 = a 113 ⋅ b 133 + a 123 ⋅ b 233 + a 133 ⋅ b 333 ;

c 231 = a 211 ⋅ b 121 + a 221 ⋅ b 221 + a 231 ⋅ b 321 ; c 232 = a 212 ⋅ b 122 + a 222 ⋅ b 222 + a 232 ⋅ b 322 ; c 233 = a 213 ⋅ b 123 + a 223 ⋅ b 223 + a 233 ⋅ b 323 ;

c 331 = a 311 ⋅ b 121 + a 321 ⋅ b 221 + a 331 ⋅ b 321 ; c 332 = a 312 ⋅ b 122 + a 322 ⋅ b 222 + a 332 ⋅ b 322 ; c 333 = a 313 ⋅ b 123 + a 323 ⋅ b 223 + a 333 ⋅ b 323 ;

It is reduce the above notes through matrix blocks

( C 3 C 2 C 1 ) = ( A 3 A 2 A 1 ) ⊗ ( B 3 B 2 B 1 ) = ( A 3 × B 3 A 2 × B 2 A 1 × B 1 )

where

A 1 = ( a 111 a 121 a 131 a 211 a 221 a 231 a 311 a 321 a 331 ) ; A 2 = ( a 112 a 122 a 132 a 212 a 222 a 232 a 312 a 322 a 332 ) ; A 3 = ( a 113 a 123 a 133 a 213 a 223 a 233 a 313 a 323 a 333 ) ; B 1 = ( b 111 b 121 b 131 b 211 b 221 b 231 b 311 b 321 b 331 ) ; B 2 = ( b 112 b 122 b 132 b 212 b 222 b 232 b 312 b 322 b 332 ) ; B 3 = ( b 113 b 123 b 133 b 213 b 223 b 233 b 313 b 323 b 333 ) ;

( C 3 C 2 C 1 ) = ( A 3 A 2 A 1 ) ⊗ ( B 3 B 2 B 1 ) = ( A 3 × B 3 A 2 × B 2 A 1 × B 1 )

and

C 1 = ( c 111 c 121 c 131 c 211 c 221 c 231 c 311 c 321 c 331 ) ; C 2 = ( c 112 c 122 c 132 c 212 c 222 c 232 c 312 c 322 c 332 ) ; C 3 = ( c 113 c 123 c 133 c 213 c 223 c 233 c 313 c 323 c 333 )

C 1 = A 1 × B 1 ; C 2 = A 2 × B 2 ; C 3 = A 3 × B 3

Remark 4.1 Two dimensional matrices can think like matrix with size m × n × 1

Easy seen from the definition 1, above it that, if a i j 2 = 0 , a i j 3 = 0 and b i j 2 = 0 , b i j 3 = 0 , ∀ i , j ∈ ( 1 , 2 , 3 ) we get, the usual 3 × 3-matrix multiplication, then will take only the first vertical layer is (or, in the language of matrix blocks would say that: A 2 = 0 ; A 3 = 0 ; B 2 = 0 ; B 3 = 0 ):

Definition 4.2. The3-D,unit matrix, associated with the “common” multiplication, must be:

I 3 × 3 × 3 = ( 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 ) thirdverticallayer thesecondverticallayer thefirstverticallayer

or, in the language of matrix blocks:

I 3 × 3 × 3 = ( I 3 × 3 I 3 × 3 I 3 × 3 )

Easy distinguish that, ∀ A ∈ M 3 × 3 × 3 ( F ) / A ⊗ I 3 × 3 × 3 = A .

Theorem 4.1 ( M 3 × 3 × 3 ( F ) , ⊗ ) is a unitary semi-Group with regard to this ordinary multiplication

Proof: 1) associative property. ∀ A , B , C ∈ M 3 × 3 × 3 ( F )

[ ( A 3 A 2 A 1 ) ⊗ ( B 3 B 2 B 1 ) ] ⊗ ( C 3 C 2 C 1 ) = ( A 3 × B 3 A 2 × B 2 A 1 × B 1 ) ⊗ ( C 3 C 2 C 1 ) = ( ( A 3 × B 3 ) × C 3 ( A 2 × B 2 ) × C 2 ( A 1 × B 1 ) × C 1 ) = ( M 3 × 3 ( F ) , × ) isasemigroup ( A 3 × ( B 3 × C 3 ) A 2 × ( B 2 × C 2 ) A 1 × ( B 1 × C 1 ) ) = ( A 3 A 2 A 1 ) ⊗ [ ( B 3 B 2 B 1 ) ⊗ ( C 3 C 2 C 1 ) ] .

2) ∃ I 3 × 3 × 3 ∈ M 3 × 3 × 3 ( F ) / ∀ A ∈ M 3 × 3 × 3 ( F ) ⇒ A × I 3 × 3 × 3 = A .

( A 3 A 2 A 1 ) ⊗ ( I 3 × 3 I 3 × 3 I 3 × 3 ) = ( A 3 × I 3 × 3 A 2 × I 3 × 3 A 1 × I 3 × 3 ) = ( M 3 × 3 ( F ) , × ) isaunitarysemigroup ( A 3 A 2 A 1 )

Theorem 4.2 ( M 3 × 3 × 3 ( F ) , + , ⊗ ) is a unitary Ring.

Proof: 1) From Theorem 2.1. ( M 3 × 3 × 3 ( F ) , + ) is abeliangrup.

2) From Theorem 4.1. ( M 3 × 3 × 3 ( F ) , ⊗ ) is a unitary semi-Group, and consequently also, ( M 3 × 3 × 3 ( F ) , ⊗ ) is a unitary semi-Group

3) ∀ A , B , C ∈ M 3 × 3 × 3 ( F ) ,

a ) A ⊗ ( B + C ) = A ⊗ B + A ⊗ C . b) ( A + B ) ⊗ C = A ⊗ C + B ⊗ C .

truly

In a similar manner proved the point (b).

Definition 5.1 The multiplication of matrix A ∈ M 3 × 3 × 3 ( F ) with scalar λ ∈ F , is matrix C = λ ∘ A ∈ M 3 × 3 × 3 ( F ) :

C = λ ∘ ( a 113 a 123 a 133 a 213 a 223 a 233 a 313 a 323 a 333 a 112 a 122 a 132 a 212 a 222 a 232 a 312 a 322 a 332 a 111 a 121 a 131 a 211 a 221 a 231 a 311 a 321 a 331 ) = ( λ ⋅ a 113 λ ⋅ a 123 λ ⋅ a 133 λ ⋅ a 213 λ ⋅ a 223 λ ⋅ a 233 λ ⋅ a 313 λ ⋅ a 323 λ ⋅ a 333 λ ⋅ a 112 λ ⋅ a 122 λ ⋅ a 132 λ ⋅ a 212 λ ⋅ a 222 λ ⋅ a 232 λ ⋅ a 312 λ ⋅ a 322 λ ⋅ a 332 λ ⋅ a 111 λ ⋅ a 121 λ ⋅ a 131 λ ⋅ a 211 λ ⋅ a 221 λ ⋅ a 231 λ ⋅ a 311 λ ⋅ a 321 λ ⋅ a 331 )

So

∘ : F × M 3 × 3 × 3 ( F ) → M 3 × 3 × 3 ( F ) ( λ , A ) ↦ λ ∘ A

Theorem 5.1 ( M 3 × 3 × 3 ( F ) , + , ∘ F ) is a vector space

Proof. is evident because, ( M 3 × 3 ( F ) , + , ∘ F ) it is the vector space, see [

Definition 5.2 The multiplication of matrix A ∈ M m × n × p ( F ) with scalar λ ∈ F , is matrix C = λ ∘ A ∈ M m × n × p ( F ) :

wherein each element of the matrix is multiplied (by multiplication of the field F) with the element λ ∈ F . Well, so we have

∘ : F × M m × n × p ( F ) → M m × n × p ( F ) ( λ , A ) ↦ λ ∘ A

Theorem 5.2 ( M m × n × p ( F ) , + , ∘ F ) is a vector space

Proof. Is evident because, ( M m × n ( F ) , + , ∘ F ) it is the vector space, see [

In this article, based on geometric considerations, and mostly considering the cube, we managed to develop the idea of the 3D matrix doing so a generalization of the 2D matrices, step by step. Furthermore, we gave a unitary ring with the elements of a field F. Initially we gave the ring 3 × 3 × 3 , 3D matrices and then generalized this concept for m × n × p , 3D matrices. At the end of this article, we present the scalar multiplication with the 3D matrices and we show that the set of 3 × 3 × 3 , 3D matrix, forms a vector space over the field F.

Zaka, O. (2017) 3D Matrix Ring with a “Common” Multipli- cation. Open Access Library Journal, 4: e3593. https://doi.org/10.4236/oalib.1103593