Multidimensional Matrix Mathematics: Algebraic Laws, Part 5 of 6

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

Abstract—This is the first series of research papers to define

multidimensional matrix mathematics, which includes

multidimensional matrix algebra and multidimensional matrix calculus. These are new branches of math created by the author with numerous applications in engineering, math, natural science, social science, and other fields. Cartesian and general tensors can be represented as multidimensional matrices or vice versa. Some Cartesian and general tensor operations can be performed as multidimensional matrix operations or vice versa. However, many aspects of multidimensional matrix math and tensor analysis are not interchangeable. Part 5 of 6 describes the commutative, associative, and distributive laws of multidimensional matrix algebra.

Index Terms—multidimensional matrix math, multidimensional matrix algebra, multidimensional matrix calculus, matrix math, matrix algebra, matrix calculus, tensor analysis

I. INTRODUCTION

Part 5 of 6 describes the commutative, associative, and distributive laws of multidimensional matrix algebra.

II.ALGEBRAIC LAWS OF MULTIDIMENSIONAL MATRIX ALGEBRA

The algebraic laws for classical matrices can be extended to multidimensional matrices. A multidimensional matrix is a concatenation of 1-D submatrices, 2-D submatrices, or both. The operations for addition, subtraction, and multiplication of multidimensional matrices are actually performed on individual 1-D submatrices and 2-D submatrices within the multidimensional matrices. The algebraic laws for classical matrices would apply to each individual 1-D or 2-D submatrix taken separately within a multidimensional matrix. If the algebraic laws for operations on classical matrices apply to each individual 1-D or 2-D submatrix within a multidimensional matrix, then they apply for operations on whole multidimensional matrices too because a multidimensional matrix consists solely of 1-D submatrices, 2-D submatrices, or both.

In the following equations, A, B, and C are multidimensional matrices and  is a scalar. Furthermore, in the following equations, multiplication refers to the multidimensional matrix product defined above.

Manuscript received March 23, 2010.

Ashu M. G. Solo is with Maverick Technologies America Inc., Suite 808, 1220 North Market Street, Wilmington, DE 19801 USA (phone: (306) 242-0566; email: [email protected]).

A.Commutative and Noncommutative Laws of Multidimensional Matrix Algebra

The commutative and noncommutative laws below are proven in the next section.

Commutative Law of Multidimensional Matrix Addition Addition of multidimensional matrices is commutative. A + B = B + A

A – B = A + (-B) = -B + A

Commutative Law of Multidimensional Matrix Multiplication by a Scalar

Multiplication of a multidimensional matrix by a scalar is commutative.

AA

Noncommutative Law of Multidimensional Matrix Subtraction

However, subtraction of multidimensional matrices is not commutative.

A – B does not have to equal B – A.

Noncommutative Law of Multidimensional Matrix Multiplication

Multiplication of multidimensional matrices is not commutative.

A *(da, db)B does not have to equal B *(da, db)A.

Noncommutative Law of Multidimensional Matrix Outer Product

Outer product of multidimensional matrices is not commutative.

AB does not have to equal BA.

B.Associative and Nonassociative Laws of Multidimensional Matrix Algebra

The associative and nonassociative laws below are proven in the next section.

Associative Law of Multidimensional Matrix Addition Addition of multidimensional matrices is associative. A + (B + C) = (A + B) + C

Associative Law of Multidimensional Matrix Multiplication

Multiplication of multidimensional matrices is associative when the first dimension and second dimension being multiplied in the first multidimensional matrix product are the same as the first dimension and second dimension being multiplied in the second multidimensional matrix product. The variable da below refers to the first dimension being multiplied for the first two multidimensional matrices. The variable db below refers to the second dimension being multiplied for the first two multidimensional matrices. Therefore, db > da.

A*(da, db) (B*(da, db)C) = (A *(da, db)B) *(da, db)C

Nonassociative Law of Multidimensional Matrix Multiplication

Multidimensional Matrix Mathematics:

Algebraic Laws, Part 5 of 6

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Multiplication of multidimensional matrices is not associative when the first dimension and second dimension being multiplied in the first multidimensional matrix product are not the same as the first dimension and second dimension being multiplied in the second multidimensional matrix product.

The variable dc below refers to the first dimension being multiplied for the second two multidimensional matrices. The variable dd below refers to the second dimension being multiplied for the second two multidimensional matrices. Therefore, dd > dc.

When da  dc or db  dd, multidimensional matrix multiplication is not associative.

A*(da, db) (B*(dc, dd)C) does not have to equal (A *(da, db)B) *(dc, dd)C whendadc or dbdd

Associative Law of Multidimensional Matrix Outer Product

The outer product of multidimensional matrices is associative.

A(BC) = (AB)C

C. Distributive Laws of Multidimensional Matrix Algebra The distributive laws below are proven in the next section. Distributive Law of Multiplication over Addition of Multidimensional Matrices

Multiplication is distributive over addition of multidimensional matrices.

A*(da, db)(B + C) = A*(da, db)B + A*(da, db)C

(A + B)*(da, db)C = A*(da, db)C + B*(da, db)C

(A + B) = A

Distributive Law of Outer Product over Addition of Multidimensional Matrices

Outer product is distributive over addition of multidimensional matrices.

A(B + C) = AB + AC (B + C)A = BA + CA

III. PROOFS OF ALGEBRAIC LAWS OF MULTIDIMENSIONAL MATRIX ALGEBRA

A.Proofs of Commutative and Noncommutative Laws of Multidimensional Matrix Algebra

Proof of Commutative Law of Multidimensional Matrix Addition

In proving the commutative property of multidimensional matrix addition, it can be proven that A + B = B + A as follows:

A = multidimensional matrix of values aijk . . . q B = multidimensional matrix of values bijk . . . q

Therefore, A + B = multidimensional matrix of values aijk . . . q + bijk . . . q.

And B + A = multidimensional matrix of values bijk . . . q+ aijk . . . q.

Because aijk . . . qand bijk . . . q are scalar values, aijk . . . q+ bijk . . . q = bijk . . . q+ aijk . . . q.

Therefore, A + B = B + A.

In proving the commutative property of multidimensional matrix addition, it can be proven that A – B = A + (-B) = -B + A as follows:

Therefore, A-B = multidimensional matrix of values aijk . . . q -bijk . . . q.

And A + (-B) =multidimensional matrix of values aijk . . . q+ (-bijk . . . q).

And -B + A =multidimensional matrix of values -bijk . . . q + aijk . . . q.

Because aijk . . . qand bijk . . . q are scalar values, aijk . . . q- bijk . . . q = aijk . . . q+ (-bijk . . . q) = -bijk . . . q + aijk . . . q.

Therefore, A – B = A + (-B) = -B + A.

Proof of Commutative Law of Multidimensional Matrix Multiplication by a Scalar

The commutative property of multidimensional matrix multiplication by a scalar, AAcan be proven as follows:

A = multidimensional matrix of values aijk . . . q

Therefore, A = multidimensional matrix of values  aijk . . . q.

And A = multidimensional matrix of values aijk . . . q * . Because aijk . . . qare scalar values,  aijk . . . q = aijk . . . q * . Therefore, AA.

Proof of Noncommutative Law of Multidimensional Matrix Subtraction

It can be proven that multidimensional matrix subtraction is not commutative. That is, A – B does not have to equal B – A. This can be provenas follows:

Therefore, A - B = multidimensional matrix of values aijk . . . q -bijk . . . q.

And B - A = multidimensional matrix of values bijk . . . q- aijk . . . q.

Because aijk . . . qand bijk . . . q are scalar values, aijk . . . q- bijk . . . q does not have to equal bijk . . . q-aijk . . . q.

Therefore, A – B does not have to equal B – A.

Proof of Noncommutative Law of Multidimensional Matrix Multiplication

It can be proven that multidimensional matrix multiplication is not commutative. That is, A *(da, db)B does not have to equal B *(da, db)A. This can be provenas follows: Let A =









1 2

3 4

       

Let B =

5 6 7 8

          _{ }  _{ }  _{ }   

A *(1, 2) B = AB =

 

17

53

       

B *(1, 2) A = BA =

5 10 6 12 21 28 24 32

       

 

 

_ _ _ _ _ _

 

Therefore, ABBA.

Therefore, A *(da, db)B does not have to equal B *(da, db)A.

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It can be proven that multidimensional matrix outer product is not commutative. That is, AB does not have to equal BA. This can be provenas follows:

Let A = multidimensional matrix of values aijk Let B = multidimensional matrix of values blmn Let C = AB

Let D = BA

Therefore, C = multidimensional matrix of values cijklmn = aijk * blmn

And D = multidimensional matrix of values dlmnijk = blmn * aijk Therefore, cijklmn does not have to equal dlmnijk.

Therefore, C does not have to equal D. Therefore, AB does not have to equal BA.

B.Proofs of Associative Laws of Multidimensional Matrix Algebra

Proof of Associative Law of Multidimensional Matrix Addition

The associative property of multidimensional matrix addition, A + (B + C) = (A + B) + C, can be proven as follows:

C = multidimensional matrix of values cijk . . . q

Therefore, A + (B + C) = multidimensional matrix of values aijk . . . q+ (bijk . . . q + cijk . . . q).

And (A + B) + C = multidimensional matrix of values (aijk . . . q + bijk . . . q) + cijk . . . q.

Because aijk . . . q, bijk . . . q, and cijk . . . q are scalar values, aijk . . . q+ (bijk . . . q + cijk . . . q) = (aijk . . . q+ bijk . . . q) + cijk . . . q.

Therefore, A + (B + C) = (A + B) + C.

Proof of Associative Law of Multidimensional Matrix Multiplication

It can be proven that multidimensional matrix multiplication is associative when the first dimension and second dimension being multiplied in the first multidimensional matrix product are the same as the first dimension and second dimension being multiplied in the second multidimensional matrix product. That is, A*(da, db) (B

*(da, db)C) = (A *(da, db)B) *(da, db)C. This can be proven as follows:

Let D = A *(da, db)B. Let E = B*(da, db)C.

Let F = A*(da, db) (B*(da, db)C)= A*(da, db)E. Let G = (A *(da, db)B) *(da, db)C = D *(da, db)C.

Therefore,

. . . where replaces index of * . . . where replaces index of

ijk q ijk q x db ijk q x da

x

d 



a b

x

e 



b c

x

f 



a e

x

g 



d c

Therefore,





... ... where replaces index of ... where replaces index of where replaces index of ... where replaces index of

2 * *

ijk q ijk q x2 db ijk q x2 da x1 db ijk q x1 da

x1 x

g 

 

a b c



ijk... whereq x2replaces index ofdb* ijk... whereq x2replaces index of da



wherex1replaces index ofdb* ijk... whereq x1replaces index of da

x1 x2 a b c



 





... where replaces index of ... where replaces index of * ... where replaces index of where replaces index of

ijk q x1 db ijk q x2 db ijk q x1 da x1 da

x1a x2b c





... where replaces index of * ... where replaces index of

ijk q x db ijk q x da

xa e





 fijk...q

Therefore, G = F.

Therefore, A*(da, db) (B*(da, db)C) = (A *(da, db)B) *(da, db)C.

Proof of Nonassociative Law of Multidimensional Matrix Multiplication

It can be proven that multidimensional matrix multiplication is not associative when the first dimension and second dimension being multiplied in the first multidimensional matrix product are not the same as the first dimension and second dimension being multiplied in the second multidimensional matrix product. That is, A*(da, db) (B

*(dc, dd)C) does not have to equal (A *(da, db)B) *(dc, dd)C when dadc or dbdd. This can be proven as follows:

Let A = B = C =

1 2 3 4 5 6 7 8

          _ _ _ _ _ _  

A*(1, 2) (B*(2, 3)C) =

143 102 293 218 637 782 861 1058

       

 

 

_ _ _ _ _ _

 

(A *(1, 2)B) *(2, 3)C =

141 166 409 490 437 518 833 1002

       

 

 

_ _  _ _  _ _ 

 

Therefore, A*(1, 2) (B*(2, 3)C) (A *(1, 2)B) *(2, 3)C.

Proof of Associative Law of Multidimensional Matrix Outer Product

The associative property of multidimensional matrix outer product, A(BC) = (AB)C, can be proven as follows:

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., qb. The indicesof multidimensional matrix C are ic, jc, kc, . . ., qc.

A = multidimensional matrix of values aia, ja, ka, . . ., qa B = multidimensional matrix of values bib, jb, kb, . . ., qb C = multidimensional matrix of values cic, jc, kc, . . ., qc Let D = AB

Let E = BC

The indicesof multidimensional matrix D are id, jd, kd, . . ., qd. The indicesof multidimensional matrix E are ie, je, ke, . . ., qe.

D = multidimensional matrix of values did, jd, kd, . . ., qd = aia, ja, ka, . . ., qa * bib, jb, kb, . . ., qb where the indices of each element did, jd, kd, . . ., qd are determined by a concatenation of the indices for aia, ja, ka, . . ., qafollowed by the indices for bib, jb, kb, . . ., qb.

E = multidimensional matrix of values eie, je, ke, . . ., qe = bib, jb, kb, . . ., qb * cic, jc, kc, . . ., qc where the indices of each element eie, je, ke, . . ., qe are determined by a concatenation of the indices for bib, jb, kb, . . ., qb followed by the indices for cic, jc, kc, . . ., qc.

Let F = A(BC)= AE. Let G = (AB)C = DC.

The indicesof multidimensional matrix F are if, jf, kf, . . ., qf. The indicesof multidimensional matrix G are ig, jg, kg, . . ., qg.

F = multidimensional matrix of values fif, jf, kf, . . ., qf = aia, ja, ka, . . ., qa * eie, je, ke, . . ., qe where the indices of each element fif, jf, kf, . . ., qf are determined by a concatenation of the indices for aia, ja, ka, . . ., qafollowed by the indices for eie, je, ke, . . ., qe.

G = multidimensional matrix of values gig, jg, kg, . . ., qg = did, jd, kd, . . ., qd * cic, jc, kc, . . ., qc where the indices of each element gig, jg, kg,

. . ., qg are determined by a concatenation of the indices for did, jd, kd, . . ., qd followed by the indices for cic, jc, kc, . . ., qc.

Therefore, F = multidimensional matrix of values fif, jf, kf, . . ., qf = aia, ja, ka, . . ., qa * bib, jb, kb, . . ., qb * cic, jc, kc, . . ., qc where the indices of each element fif, jf, kf, . . ., qf are determined by a concatenation of the indices for aia, ja, ka, . . ., qafollowed by the indices for bib, jb, kb, . . ., qband then cic, jc, kc, . . ., qc.

And G = multidimensional matrix of values gig, jg, kg, . . ., qg = aia, ja, ka, . . ., qa * bib, jb, kb, . . ., qb* cic, jc, kc, . . ., qc where the indices of each element gig, jg, kg, . . ., qg are determined by a concatenation of the indices for aia, ja, ka, . . ., qafollowed by the indices for bib, jb, kb, . . ., qb and then cic, jc, kc, . . ., qc.

Therefore, F = G.

Therefore, A(BC) = (AB)C.

C.Proofs of Distributive Laws of Multidimensional Matrix Algebra

Proof of Distributive Law of Multiplication over Addition of Multidimensional Matrices

In proving that multiplication is distributive over addition of multidimensional matrices, it can be proven that A*(da, db)

(B + C) = A*(da, db)B + A*(da, db)C as follows: Let M1 = B + C

Let M2= A*(da, db)M1 = A*(da, db)(B + C) Let N1 = A*(da, db)B

Let N2 = A*(da, db)C

Let N3 = N1 + N2 = A*(da, db)B + A*(da, db)C

m1ijk . . . q= bijk . . . q+ cijk . . . q

m2ijk . . . q=



_xaijk...qwherexreplaces index ofdb*m1ijk...qwherexreplaces index of da





... where replaces index of * ... where replaces index of ... where replaces index of

ijk q x db ijk q x da ijk q x da

xa b c







n1ijk . . . q= ijk...qwherexreplaces index ofdb* ijk...qwherexreplaces index of da

xa b



n2ijk . . . q= ijk...qwherexreplaces index ofdb* ijk...qwherexreplaces index of da

xa c



n3ijk . . . q= n1ijk . . . q+ n2ijk . . . q

. . . where replaces index of . . . where replaces index of

. . . where replaces index of . . . where replaces index of *

*

x

a b

a c

 







... where replaces index of * ... where replaces index of ... where replaces index of

ijk q x db ijk q x da ijk q x da

xa b c







Therefore, m2ijk . . . q = n3ijk . . . q Therefore, M2 = N3

Therefore, A*(da, db)(B + C) = A*(da, db)B + A*(da, db)C In proving that multiplication is distributive over addition of multidimensional matrices, it can be proven that (A + B) *(da, db)C = A*(da, db)C + B*(da, db)C as follows:

Let M1 = A + B

Let M2 = M1*(da, db)C = (A + B)*(da, db)C Let N1 = A*(da, db)C

Let N2 = B*(da, db)C

Let N3 = N1 + N2 = A*(da, db)C + B*(da, db)C

m1ijk . . . q= aijk . . . q+ bijk . . . q

m2ijk . . . q =



_xm1ijk...qwherexreplaces index ofdb*cijk...qwherexreplaces index of da



ijk... whereq xreplaces index of db ijk... whereq xreplaces index of db



* ijk... whereq xreplaces index ofda

x a b c







n1ijk . . . q =



_xaijk...qwherexreplaces index ofdb*cijk...qwherexreplaces index of da n2ijk . . . q = ijk...qwherexreplaces index ofdb* ijk...qwherexreplaces index of da

xb c



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... where replaces index of * ... where replaces index of

xa c





ijk... whereq xreplaces index ofdb* ijk... whereq xreplaces index of da

xb c







ijk... whereq xreplaces index of db ijk... whereq xreplaces index of db



* ijk... whereq xreplaces index ofda

x a b c







Therefore, m2ijk . . . q = n3ijk . . . q Therefore, M2 = N3

Therefore, (A + B)*(da, db)C = A*(da, db)C + B*(da, db)C It can be proven that multiplication by a scalar is distributive over addition of multidimensional matrices, (A + B) = A as follows:

Then A + B = multidimensional matrix of values aijk . . . q + bijk . . . q

Therefore, (A + B) = multidimensional matrix of values aijk . . . q + bijk . . . q

Then A = multidimensional matrix of values aijk . . . q And B = multidimensional matrix of values bijk . . . q Therefore, A + B = multidimensional matrix of values aijk . . . qbijk . . . q

Therefore,(A + B) = A

Proof of Distributive Law of Outer Product over Addition of Multidimensional Matrices

In proving that outer product is distributive over addition of multidimensional matrices, it can be proven that A(B + C) = AB + AC as follows:

The indices of multidimensional matrix A are ia, ja, ka, . . . qa. The indicesof multidimensional matrix B are ib, jb, kb, . . . qb. The indicesof multidimensional matrix C are ic, jc, kc, . . . qc.

A = multidimensional matrix of values aia, ja, ka, . . . qa B = multidimensional matrix of values bib, jb, kb, . . . qb C = multidimensional matrix of values cic, jc, kc, . . . qc Let M1 = B + C

Let M2 = AM1 = A(B + C)

The indices of multidimensional matrix M1 are im1, jm1, km1, . . . qm1. The indicesof multidimensional matrix M2 are im2, jm2, km2, . . . qm2.

Therefore, M1 = multidimensional matrix of values m1im1, jm1, km1, . . . qm1 = bib, jb, kb, . . . qb+ cic, jc, kc, . . . qc.

Therefore, M2 = multidimensional matrix of values m2im2, jm2, km2, . . . qm2 = aia, ja, ka, . . . qa * m1im1, jm1, km1, . . . qm1 where the indices of each element m2im2, jm2, km2, . . . qm2 are determined by a concatenation of the indices for aia, ja, ka, . . . qafollowed by the indices for m1im1, jm1, km1, . . . qm1.

Therefore, M2 = multidimensional matrix of values m2im2, jm2, km2, . . . qm2 = aia, ja, ka, . . . qa * (bib, jb, kb, . . . qb+ cic, jc, kc, . . . qc) where the indices of each element m2im2, jm2, km2, . . . qm2 are determined by a concatenation of the indices for aia, ja, ka, . . . qa followed by the indices for bib, jb, kb, . . . qbor cic, jc, kc, . . . qc.

Because aia, ja, ka, . . . qa, bib, jb, kb, . . . qb, and cic, jc, kc, . . . qc are scalar values, m2im2, jm2, km2, . . . qm2 = aia, ja, ka, . . . qa, * (bib, jb, kb, . . . qb + cic, jc, kc, . . . qc) = aia, ja, ka, . . . qa * bib, jb, kb, . . . qb + aia, ja, ka, . . . qa * cic, jc, kc, . . . qc.

Therefore, M2 = multidimensional matrix of values m2im2, jm2, km2, . . . qm2 = aia, ja, ka, . . . qa * bib, jb, kb, . . . qb + aia, ja, ka, . . . qa * cic, jc, kc, . . . qc where the indices of each element m2im2, jm2, km2, . . . qm2 are determined by a concatenation of the indices for aia, ja, ka, . . . qa followed by the indices for bib, jb, kb, . . . qb orare determined by a concatenation of the indices for aia, ja, ka, . . . qafollowed by the indices for cic, jc, kc, . . . qc.

Let N1 = AB Let N2 = AC

Let N3 = N1 + N2 = AB + AC

The indices of multidimensional matrix N1 are in1, jn1, kn1, . . . qn1. The indicesof multidimensional matrix N2 are in2, jn2, kn2, . . . qn2. The indicesof multidimensional matrix N3 are in3, jn3, kn3, . . . qn3.

Therefore, N1 = multidimensional matrix of values n1in1, jn1, kn1, . . . qn1 = aia, ja, ka, . . . qa * bib, jb, kb, . . . qb where the indices of each element n1in1, jn1, kn1, . . . qn1 are determined by a concatenation of the indices for aia, ja, ka, . . . qafollowed by the indices for bib, jb, kb, . . . qb.

Therefore, N2 = multidimensional matrix of values n2in2, jn2, kn2, . . . qn2 = aia, ja, ka, . . . qa * cic, jc, kc, . . . qc where the indices of each element n2in2, jn2, kn2, . . . qn2 are determined by a concatenation of the indices for aia, ja, ka, . . . qafollowed by the indices for cic, jc, kc, . . . qc.

Therefore, N3 = multidimensional matrix of values n3in3, jn3, kn3, . . . qn3 = aia, ja, ka, . . . qa * bib, jb, kb, . . . qb + aia, ja, ka, . . . qa * cic, jc, kc, . . . qc where the indices of each element n3in3, jn3, kn3, . . . qn3 are determined by a concatenation of the indices for aia, ja, ka, . . . qa followed by the indices for bib, jb, kb, . . . qb orare determined by a concatenation of the indices for aia, ja, ka, . . . qafollowed by the indices for cic, jc, kc, . . . qc.

Therefore, M2 = N3.

Therefore, A(B + C) = AB + AC.

In proving that outer product is distributive over addition of multidimensional matrices, it can be proven that (B + C)A = BA + CA as follows:

The indices of multidimensional matrix A are ia, ja, ka, . . . qa. The indicesof multidimensional matrix B are ib, jb, kb, . . . qb. The indicesof multidimensional matrix C are ic, jc, kc, . . . qc.

A = multidimensional matrix of values aia, ja, ka, . . . qa B = multidimensional matrix of values bib, jb, kb, . . . qb C = multidimensional matrix of values cic, jc, kc, . . . qc Let M1 = B + C

Let M2 = M1A = (B + C)A

The indices of multidimensional matrix M1 are im1, jm1, km1, . . . qm1. The indicesof multidimensional matrix M2 are im2, jm2, km2, . . . qm2.

Therefore, M1 = multidimensional matrix of values m1im1, jm1, km1, . . . qm1 = bib, jb, kb, . . . qb+ cic, jc, kc, . . . qc.

Therefore, M2 = multidimensional matrix of values m2im2, jm2, km2, . . . qm2 = m1im1, jm1, km1, . . . qm1* aia, ja, ka, . . . qawhere the indices of each element m2im2, jm2, km2, . . . qm2 are determined by a concatenation of the indices for m1im1, jm1, km1, . . . qm1 followed by the indices for aia, ja, ka, . . . qa.

Therefore, M2 = multidimensional matrix of values m2im2, jm2, km2, . . . qm2 = (bib, jb, kb, . . . qb+ cic, jc, kc, . . . qc) * aia, ja, ka, . . . qa where the indices of each element m2im2, jm2, km2, . . . qm2 are determined by a concatenation of the indices for bib, jb, kb, . . . qbor cic, jc, kc, . . . qc followed bythe indices for aia, ja, ka, . . . qa.

(6)

Therefore, M2 = multidimensional matrix of values m2im2, jm2, km2, . . . qm2 = bib, jb, kb, . . . qb * aia, ja, ka, . . . qa + cic, jc, kc, . . . qc* aia, ja, ka, . . . qa where the indices of each element m2im2, jm2, km2, . . . qm2 are determined by a concatenation of the indices for bib, jb, kb, . . . qb followed by the indices for aia, ja, ka, . . . qa orare determined by a concatenation of the indices for cic, jc, kc, . . . qc followed by the indices for aia, ja, ka, . . . qa.

Let N1 = BA Let N2 = CA

Let N3 = N1 + N2 = BA + CA

The indices of multidimensional matrix N1 are in1, jn1, kn1, . . . qn1. The indicesof multidimensional matrix N2 are in2, jn2, kn2, . . . qn2. The indicesof multidimensional matrix N3 are in3, jn3, kn3, . . . qn3.

Therefore, N1 = multidimensional matrix of values n1in1, jn1, kn1, . . . qn1 = bib, jb, kb, . . . qb * aia, ja, ka, . . . qa where the indices of each element n1in1, jn1, kn1, . . . qn1 are determined by a concatenation of the indices for bib, jb, kb, . . . qbfollowed by the indices for aia, ja, ka, . . . qa.

Therefore, N2 = multidimensional matrix of values n2in2, jn2, kn2, . . . qn2 = cic, jc, kc, . . . qc* aia, ja, ka, . . . qa where the indices of each element n2in2, jn2, kn2, . . . qn2 are determined by a concatenation of the indices for cic, jc, kc, . . . qc followed by the indices for aia, ja, ka, . . . qa.

Therefore, N3 = multidimensional matrix of values n3in3, jn3, kn3, . . . qn3 = bib, jb, kb, . . . qb * aia, ja, ka, . . . qa + cic, jc, kc, . . . qc* aia, ja, ka, . . . qa where the indices of each element n3in3, jn3, kn3, . . . qn3 are determined by a concatenation of the indices for bib, jb, kb, . . . qb followed by the indices for aia, ja, ka, . . . qa orare determined by a concatenation of the indices for cic, jc, kc, . . . qc followed by the indices for aia, ja, ka, . . . qa.

Therefore, M2 = N3.

Therefore, (B + C)A = BA + CA.

IV. CONCLUSION

Part 5 of 6 described the commutative, associative, and distributive laws of multidimensional matrix algebra.

Part 6 of 6 describes the solution of systems of linear equations using multidimensional matrices. Also, part 6 of 6 defines the multidimensional matrix calculus operations for differentiation and integration.

REFERENCES

[1] Franklin, Joel L. [2000] Matrix Theory. Mineola, N.Y.: Dover. [2] Young, Eutiquio C. [1992] Vector and Tensor Analysis. 2d ed. Boca