Proceedings of the World Congress on Engineering 2010 Vol III WCE 2010, June 30 - July 2, 2010, London, U.K.
Multidimensional Matrix Mathematics: Algebraic Laws, Part 5 of 6
Ashu M. G. Solo
A. Commutative and Noncommutative Laws of Abstract—This is the first series of research papers to define multidimensional matrix mathematics, which includes Multidimensional Matrix Algebra multidimensional matrix algebra and multidimensional matrix The commutative and noncommutative laws below are calculus. These are new branches of math created by the author proven in the next section. with numerous applications in engineering, math, natural Commutative Law of Multidimensional Matrix Addition science, social science, and other fields. Cartesian and general Addition of multidimensional matrices is commutative. tensors can be represented as multidimensional matrices or vice A + B = B + A versa. Some Cartesian and general tensor operations can be A – B = A + (-B) = -B + A performed as multidimensional matrix operations or vice versa. However, many aspects of multidimensional matrix math and Commutative Law of Multidimensional Matrix tensor analysis are not interchangeable. Part 5 of 6 describes Multiplication by a Scalar the commutative, associative, and distributive laws of Multiplication of a multidimensional matrix by a scalar is multidimensional matrix algebra. commutative. AA Index Terms—multidimensional matrix math, Noncommutative Law of Multidimensional Matrix multidimensional matrix algebra, multidimensional matrix Subtraction calculus, matrix math, matrix algebra, matrix calculus, tensor However, subtraction of multidimensional matrices is not analysis commutative.
A – B does not have to equal B – A. Noncommutative Law of Multidimensional Matrix I. INTRODUCTION Multiplication Part 5 of 6 describes the commutative, associative, and Multiplication of multidimensional matrices is not distributive laws of multidimensional matrix algebra. commutative. A *(da, db) B does not have to equal B *(da, db) A. Noncommutative Law of Multidimensional Matrix Outer II. ALGEBRAIC LAWS OF MULTIDIMENSIONAL MATRIX Product ALGEBRA Outer product of multidimensional matrices is not The algebraic laws for classical matrices can be extended to commutative. multidimensional matrices. A multidimensional matrix is a A B does not have to equal B A. concatenation of 1-D submatrices, 2-D submatrices, or both. B. Associative and Nonassociative Laws of Multidimensional The operations for addition, subtraction, and multiplication of Matrix Algebra multidimensional matrices are actually performed on The associative and nonassociative laws below are proven individual 1-D submatrices and 2-D submatrices within the in the next section. multidimensional matrices. The algebraic laws for classical Associative Law of Multidimensional Matrix Addition matrices would apply to each individual 1-D or 2-D submatrix Addition of multidimensional matrices is associative. taken separately within a multidimensional matrix. If the A + (B + C) = (A + B) + C algebraic laws for operations on classical matrices apply to Associative Law of Multidimensional Matrix each individual 1-D or 2-D submatrix within a Multiplication multidimensional matrix, then they apply for operations on Multiplication of multidimensional matrices is associative whole multidimensional matrices too because a when the first dimension and second dimension being multidimensional matrix consists solely of 1-D submatrices, multiplied in the first multidimensional matrix product are the 2-D submatrices, or both. same as the first dimension and second dimension being In the following equations, A, B, and C are multiplied in the second multidimensional matrix product. multidimensional matrices and is a scalar. Furthermore, in The variable da below refers to the first dimension being the following equations, multiplication refers to the multiplied for the first two multidimensional matrices. The multidimensional matrix product defined above. variable db below refers to the second dimension being multiplied for the first two multidimensional matrices. Manuscript received March 23, 2010. Therefore, db > da. Ashu M. G. Solo is with Maverick Technologies America Inc., Suite 808, A *(da, db) (B *(da, db) C) = (A *(da, db) B) *(da, db) C 1220 North Market Street, Wilmington, DE 19801 USA (phone: (306) Nonassociative Law of Multidimensional Matrix 242-0566; email: [email protected]). Multiplication
ISBN: 978-988-18210-8-9 WCE 2010 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering 2010 Vol III WCE 2010, June 30 - July 2, 2010, London, U.K.
Multiplication of multidimensional matrices is not Therefore, A - B = multidimensional matrix of values aijk . . . q - associative when the first dimension and second dimension bijk . . . q. being multiplied in the first multidimensional matrix product And A + (-B) = multidimensional matrix of values aijk . . . q + are not the same as the first dimension and second dimension (-bijk . . . q). being multiplied in the second multidimensional matrix And -B + A = multidimensional matrix of values -bijk . . . q + aijk product. . . . q. The variable dc below refers to the first dimension being Because aijk . . . q and bijk . . . q are scalar values, aijk . . . q - bijk . . . q = multiplied for the second two multidimensional matrices. The aijk . . . q + (-bijk . . . q) = -bijk . . . q + aijk . . . q. variable dd below refers to the second dimension being Therefore, A – B = A + (-B) = -B + A. multiplied for the second two multidimensional matrices. Proof of Commutative Law of Multidimensional Matrix Therefore, dd > dc. Multiplication by a Scalar When da dc or db dd, multidimensional matrix The commutative property of multidimensional matrix multiplication is not associative. multiplication by a scalar, AAcan be proven as A *(da, db) (B *(dc, dd) C) does not have to equal (A *(da, db) B) *(dc, follows: dd) C when da dc or db dd A = multidimensional matrix of values aijk . . . q Associative Law of Multidimensional Matrix Outer Therefore, A = multidimensional matrix of values aijk . . . Product q. The outer product of multidimensional matrices is And A = multidimensional matrix of values aijk . . . q * . associative. Because aijk . . . q are scalar values, aijk . . . q = aijk . . . q * . A (B C) = (A B) C Therefore, AA. C. Distributive Laws of Multidimensional Matrix Algebra Proof of Noncommutative Law of Multidimensional Matrix Subtraction The distributive laws below are proven in the next section. It can be proven that multidimensional matrix subtraction is Distributive Law of Multiplication over Addition of not commutative. That is, A – B does not have to equal B – A. Multidimensional Matrices This can be proven as follows: Multiplication is distributive over addition of A = multidimensional matrix of values a multidimensional matrices. ijk . . . q B = multidimensional matrix of values b A * (B + C) = A * B + A * C ijk . . . q (da, db) (da, db) (da, db) Therefore, A - B = multidimensional matrix of values a - (A + B) * C = A * C + B * C ijk . . . q (da, db) (da, db) (da, db) b . (A + B) = A ijk . . . q And B - A = multidimensional matrix of values b - a Distributive Law of Outer Product over Addition of ijk . . . q ijk . . . Multidimensional Matrices . q Because a and b are scalar values, a - b Outer product is distributive over addition of ijk . . . q ijk . . . q ijk . . . q ijk . . . q does not have to equal b - a . multidimensional matrices. ijk . . . q ijk . . . q Therefore, A – B does not have to equal B – A. A (B + C) = A B + A C Proof of Noncommutative Law of Multidimensional (B + C) A = B A + C A Matrix Multiplication It can be proven that multidimensional matrix multiplication is not commutative. That is, A *(da, db) B does III. PROOFS OF ALGEBRAIC LAWS OF MULTIDIMENSIONAL not have to equal B *(da, db) A. This can be proven as follows: MATRIX ALGEBRA 12 Let A = A. Proofs of Commutative and Noncommutative Laws of 34 Multidimensional Matrix Algebra Proof of Commutative Law of Multidimensional Matrix 5 Addition 6 Let B = In proving the commutative property of multidimensional 7 matrix addition, it can be proven that A + B = B + A as follows: 8 A = multidimensional matrix of values aijk . . . q 17 B = multidimensional matrix of values bijk . . . q A *(1, 2) B = AB = 53 Therefore, A + B = multidimensional matrix of values aijk . . . q + bijk . . . q. 5 10 And B + A = multidimensional matrix of values b + a ijk . . . q ijk . 6 12 . . . q B *(1, 2) A = BA = Because a and b are scalar values, a + b 21 28 ijk . . . q ijk . . . q ijk . . . q ijk . . . q = b + a . ijk . . . q ijk . . . q 24 32 Therefore, A + B = B + A. In proving the commutative property of multidimensional Therefore, AB BA. matrix addition, it can be proven that A – B = A + (-B) = -B + Therefore, A *(da, db) B does not have to equal B *(da, db) A. A as follows: Proof of Noncommutative Law of Multidimensional Matrix Outer Product A = multidimensional matrix of values aijk . . . q B = multidimensional matrix of values bijk . . . q
ISBN: 978-988-18210-8-9 WCE 2010 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering 2010 Vol III WCE 2010, June 30 - July 2, 2010, London, U.K.
It can be proven that multidimensional matrix outer product C = multidimensional matrix of values cijk . . . q is not commutative. That is, A B does not have to equal Therefore, A + (B + C) = multidimensional matrix of values B A. This can be proven as follows: aijk . . . q + (bijk . . . q + cijk . . . q).
Let A = multidimensional matrix of values aijk And (A + B) + C = multidimensional matrix of values (aijk . . . q Let B = multidimensional matrix of values blmn + bijk . . . q) + cijk . . . q. Let C = A B Because aijk . . . q, bijk . . . q, and cijk . . . q are scalar values, aijk . . . q + Let D = B A (bijk . . . q + cijk . . . q) = (aijk . . . q + bijk . . . q) + cijk . . . q. Therefore, A + (B + C) = (A + B) + C. Therefore, C = multidimensional matrix of values cijklmn = aijk * Proof of Associative Law of Multidimensional Matrix blmn Multiplication And D = multidimensional matrix of values dlmnijk = blmn * aijk It can be proven that multidimensional matrix Therefore, cijklmn does not have to equal dlmnijk. Therefore, C does not have to equal D. multiplication is associative when the first dimension and Therefore, A B does not have to equal B A. second dimension being multiplied in the first multidimensional matrix product are the same as the first B. Proofs of Associative Laws of Multidimensional Matrix dimension and second dimension being multiplied in the Algebra second multidimensional matrix product. That is, A *(da, db) (B Proof of Associative Law of Multidimensional Matrix *(da, db) C) = (A *(da, db) B) *(da, db) C. This can be proven as Addition follows: The associative property of multidimensional matrix Let D = A *(da, db) B. addition, A + (B + C) = (A + B) + C, can be proven as Let E = B *(da, db) C. follows: Let F = A *(da, db) (B *(da, db) C) = A *(da, db) E. A = multidimensional matrix of values aijk . . . q Let G = (A *(da, db) B) *(da, db) C = D *(da, db) C. B = multidimensional matrix of values bijk . . . q
Therefore, dijk. . . q a ijk . . . q where x replaces index of db* b ijk . . . q where x replaces index of da x eijk. . . q b ijk . . . q where x replaces index of db* c ijk . . . q where x replaces index of da x fijk. . . q a ijk . . . q where x replaces index of db* e ijk . . . q where x replaces index of da x gijk. . . q d ijk . . . q where x replaces index of db* c ijk . . . q where x replaces index of da x Therefore,
gijk... q a ijk ... q where x2 replaces index of db** b ijk ... q where x2 replaces index of da where x1 replaces index of db c ijk ... q where x1 replaces index of da x1 x2
aijk... q where x2 replaces index of db** b ijk ... q where x2 replaces index of da where x1 replaces index of db c ijk ... q where x1 replaces index of da x1 x2
aijk... q where x1 replaces index of db b ijk ... q where x2 replaces index of db* c ijk ... q where x1 replaces index of da where x1 replaces index of da x1 x2
aeijk... q where x replaces index of db * ijk ... q where x replaces index of da fijk... q x 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 143 102 Multiplication 293 218 It can be proven that multidimensional matrix A *(1, 2) (B *(2, 3) C) = 637 782 multiplication is not associative when the first dimension and second dimension being multiplied in the first 861 1058 multidimensional matrix product are not the same as the first dimension and second dimension being multiplied in the 141 166 second multidimensional matrix product. That is, A *(da, db) (B 409 490 * C) does not have to equal (A * B) * C when (A *(1, 2) B) *(2, 3) C = (dc, dd) (da, db) (dc, dd) 437 518 da dc or db dd. This can be proven as follows: 833 1002 12 Therefore, A * (B * C) (A * B) * C. 34 (1, 2) (2, 3) (1, 2) (2, 3) Let A = B = C = Proof of Associative Law of Multidimensional Matrix 56 Outer Product The associative property of multidimensional matrix outer 78 product, A (B C) = (A B) C, can be proven as follows: The indices of multidimensional matrix A are ia, ja, ka, . . ., qa. The indices of multidimensional matrix B are ib, jb, kb, . .
ISBN: 978-988-18210-8-9 WCE 2010 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering 2010 Vol III WCE 2010, June 30 - July 2, 2010, London, U.K.
., qb. The indices of multidimensional matrix C are ic, jc, kc, . . ., qg are determined by a concatenation of the indices for did, jd, . . ., qc. kd, . . ., qd followed by the indices for cic, jc, kc, . . ., qc. A = multidimensional matrix of values aia, ja, ka, . . ., qa Therefore, F = multidimensional matrix of values fif, jf, kf, . . ., qf = B = multidimensional matrix of values bib, jb, kb, . . ., qb aia, ja, ka, . . ., qa * bib, jb, kb, . . ., qb * cic, jc, kc, . . ., qc where the indices of C = multidimensional matrix of values cic, jc, kc, . . ., qc each element fif, jf, kf, . . ., qf are determined by a concatenation of Let D = A B the indices for aia, ja, ka, . . ., qa followed by the indices for bib, jb, kb, Let E = B C . . ., qb and then cic, jc, kc, . . ., qc.
The indices of multidimensional matrix D are id, jd, kd, . . ., And G = multidimensional matrix of values gig, jg, kg, . . ., qg = aia, qd. The indices of multidimensional matrix E are ie, je, ke, . . ja, ka, . . ., qa * bib, jb, kb, . . ., qb * cic, jc, kc, . . ., qc where the indices of ., qe. each element gig, jg, kg, . . ., qg are determined by a concatenation D = multidimensional matrix of values did, jd, kd, . . ., qd = aia, ja, ka, of the indices for aia, ja, ka, . . ., qa followed by the indices for bib, . . ., qa * bib, jb, kb, . . ., qb where the indices of each element did, jd, kd, jb, kb, . . ., qb and then cic, jc, kc, . . ., qc. . . ., qd are determined by a concatenation of the indices for aia, ja, Therefore, F = G. ka, . . ., qa followed by the indices for bib, jb, kb, . . ., qb. Therefore, A (B C) = (A B) C. E = multidimensional matrix of values eie, je, ke, . . ., qe = bib, jb, kb, C. Proofs of Distributive Laws of Multidimensional Matrix . . ., qb * cic, jc, kc, . . ., qc where the indices of each element eie, je, ke, Algebra . . ., qe are determined by a concatenation of the indices for bib, jb, Proof of Distributive Law of Multiplication over Addition kb, . . ., qb followed by the indices for cic, jc, kc, . . ., qc. Let F = A (B C) = A E. of Multidimensional Matrices In proving that multiplication is distributive over addition Let G = (A B) C = D C. of multidimensional matrices, it can be proven that A * The indices of multidimensional matrix F are if, jf, kf, . . ., qf. (da, db) (B + C) = A * B + A * C as follows: The indices of multidimensional matrix G are ig, jg, kg, . . ., (da, db) (da, db) Let M1 = B + C qg. Let M2 = A *(da, db) M1 = A *(da, db) (B + C) F = multidimensional matrix of values fif, jf, kf, . . ., qf = aia, ja, ka, . . Let N1 = A *(da, db) B ., qa * eie, je, ke, . . ., qe where the indices of each element fif, jf, kf, . . ., Let N2 = A *(da, db) C qf are determined by a concatenation of the indices for aia, ja, ka, Let N3 = N1 + N2 = A *(da, db) B + A *(da, db) C . . ., qa followed 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,
m1ijk . . . q = bijk . . . q + cijk . . . q
m2ijk . . . q = aijk... q where x replaces index of db* m1 ijk ... q where x replaces index of da x aijk... q where x replaces index of db* b ijk ... q where x replaces index of da c ijk ... q where x replaces index of da x
n1ijk . . . q = abijk... q where x replaces index of db* ijk ... q where x replaces index of da x
n2ijk . . . q = acijk... q where x replaces index of db* ijk ... q where x replaces index of da x
n3ijk . . . q = n1ijk . . . q + n2ijk . . . q abijk... q where x replaces index of db* ijk ... q where x replaces index of da x acijk... q where x replaces index of db* ijk ... q where x replaces index of da x aijk... q where x replaces index of db* b ijk ... q where x replaces index of da c ijk ... q where x replaces index of da x
Therefore, m2ijk . . . q = n3ijk . . . q Let M1 = A + B Therefore, M2 = N3 Let M2 = M1 *(da, db) C = (A + B) *(da, db) C Therefore, A *(da, db) (B + C) = A *(da, db) B + A *(da, db) C Let N1 = A *(da, db) C In proving that multiplication is distributive over addition Let N2 = B *(da, db) C of multidimensional matrices, it can be proven that (A + B) Let N3 = N1 + N2 = A *(da, db) C + B *(da, db) C *(da, db) C = A *(da, db) C + B *(da, db) C as follows:
m1ijk . . . q = aijk . . . q + bijk . . . q
m2ijk . . . q = m1ijk... q where x replaces index of db* c ijk ... q where x replaces index of da x aijk... q where x replaces index of db b ijk ... q where x replaces index of db* c ijk ... q where x replaces index of da x
n1ijk . . . q = acijk... q where x replaces index of db* ijk ... q where x replaces index of da x
n2ijk . . . q = bcijk... q where x replaces index of db* ijk ... q where x replaces index of da x
n3ijk . . . q = n1ijk . . . q + n2ijk . . . q
ISBN: 978-988-18210-8-9 WCE 2010 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering 2010 Vol III WCE 2010, June 30 - July 2, 2010, London, U.K.
acijk... q where x replaces index of db* ijk ... q where x replaces index of da bcijk... q where x replaces index of db* ijk ... q where x replaces index of da x x aijk... q where x replaces index of db b ijk ... q where x replaces index of db* c ijk ... q where x replaces index of da x
Therefore, m2ijk . . . q = n3ijk . . . q Let N1 = A B Therefore, M2 = N3 Let N2 = A C Therefore, (A + B) *(da, db) C = A *(da, db) C + B *(da, db) C Let N3 = N1 + N2 = A B + A C It can be proven that multiplication by a scalar is The indices of multidimensional matrix N1 are in1, jn1, kn1, . distributive over addition of multidimensional matrices, (A . . qn1. The indices of multidimensional matrix N2 are in2, + B) = A as follows: jn2, kn2, . . . qn2. The indices of multidimensional matrix N3 A = multidimensional matrix of values aijk . . . q are in3, jn3, kn3, . . . qn3. B = multidimensional matrix of values bijk . . . q Therefore, N1 = multidimensional matrix of values n1in1, jn1, Then A + B = multidimensional matrix of values aijk . . . q + bijk kn1, . . . qn1 = aia, ja, ka, . . . qa * bib, jb, kb, . . . qb where the indices of each . . . q element n1in1, jn1, kn1, . . . qn1 are determined by a concatenation of Therefore, (A + B) = multidimensional matrix of values aijk the indices for aia, ja, ka, . . . qa followed by the indices for bib, jb, kb, . . . q + bijk . . . q . . . qb. Then A = multidimensional matrix of values aijk . . . q Therefore, N2 = multidimensional matrix of values n2in2, jn2, And B = multidimensional matrix of values bijk . . . q kn2, . . . qn2 = aia, ja, ka, . . . qa * cic, jc, kc, . . . qc where the indices of each Therefore, A + B = multidimensional matrix of values aijk element n2in2, jn2, kn2, . . . qn2 are determined by a concatenation of the indices for a followed by the indices for c . . . qbijk . . . q ia, ja, ka, . . . qa ic, jc, kc, Therefore,(A + B) = A . . . qc. Proof of Distributive Law of Outer Product over Therefore, N3 = multidimensional matrix of values n3in3, jn3, Addition of Multidimensional Matrices kn3, . . . qn3 = aia, ja, ka, . . . qa * bib, jb, kb, . . . qb + aia, ja, ka, . . . qa * cic, jc, kc, In proving that outer product is distributive over addition of . . . qc where the indices of each element n3in3, jn3, kn3, . . . qn3 are multidimensional matrices, it can be proven that A (B + C) determined by a concatenation of the indices for aia, ja, ka, . . . qa = A B + A C as follows: followed by the indices for bib, jb, kb, . . . qb or are determined by a The indices of multidimensional matrix A are ia, ja, ka, . . . concatenation of the indices for aia, ja, ka, . . . qa followed by the qa. The indices of multidimensional matrix B are ib, jb, kb, . . indices for cic, jc, kc, . . . qc. Therefore, M2 = N3. . qb. The indices of multidimensional matrix C are ic, jc, kc, . . . qc. Therefore, A (B + C) = A B + A C. In proving that outer product is distributive over addition of A = multidimensional matrix of values aia, ja, ka, . . . qa multidimensional matrices, it can be proven that (B + C) A B = multidimensional matrix of values bib, jb, kb, . . . qb = B A + C A as follows: C = multidimensional matrix of values cic, jc, kc, . . . qc Let M1 = B + C The indices of multidimensional matrix A are ia, ja, ka, . . . Let M2 = A M1 = A (B + C) qa. The indices of multidimensional matrix B are ib, jb, kb, . . The indices of multidimensional matrix M1 are im1, jm1, . qb. The indices of multidimensional matrix C are ic, jc, kc, . . . qc. km1, . . . qm1. The indices of multidimensional matrix M2 are im2, jm2, km2, . . . qm2. A = multidimensional matrix of values aia, ja, ka, . . . qa B = multidimensional matrix of values bib, jb, kb, . . . qb Therefore, M1 = multidimensional matrix of values m1im1, jm1, C = multidimensional matrix of values c km1, . . . qm1 = bib, jb, kb, . . . qb + cic, jc, kc, . . . qc. ic, jc, kc, . . . qc Let M1 = B + C Therefore, M2 = multidimensional matrix of values m2im2, jm2, Let M2 = M1 A = (B + C) A km2, . . . qm2 = aia, ja, ka, . . . qa * m1im1, jm1, km1, . . . qm1 where the indices of each element m2im2, jm2, km2, . . . qm2 are determined by a The indices of multidimensional matrix M1 are im1, jm1, concatenation of the indices for aia, ja, ka, . . . qa followed by the km1, . . . qm1. The indices of multidimensional matrix M2 are indices for m1im1, jm1, km1, . . . qm1. im2, jm2, km2, . . . qm2. Therefore, M2 = multidimensional matrix of values m2im2, jm2, Therefore, M1 = multidimensional matrix of values m1im1, jm1, km2, . . . qm2 = aia, ja, ka, . . . qa * (bib, jb, kb, . . . qb + cic, jc, kc, . . . qc) where km1, . . . qm1 = bib, jb, kb, . . . qb + cic, jc, kc, . . . qc. the indices of each element m2im2, jm2, km2, . . . qm2 are determined Therefore, M2 = multidimensional matrix of values m2im2, jm2, by a concatenation of the indices for aia, ja, ka, . . . qa followed by km2, . . . qm2 = m1im1, jm1, km1, . . . qm1 * aia, ja, ka, . . . qa where the indices the indices for bib, jb, kb, . . . qb or cic, jc, kc, . . . qc. of each element m2im2, jm2, km2, . . . qm2 are determined by a Because aia, ja, ka, . . . qa, bib, jb, kb, . . . qb, and cic, jc, kc, . . . qc are scalar concatenation of the indices for m1im1, jm1, km1, . . . qm1 followed values, m2im2, jm2, km2, . . . qm2 = aia, ja, ka, . . . qa, * (bib, jb, kb, . . . qb + cic, by the indices for aia, ja, ka, . . . qa. jc, kc, . . . qc) = aia, ja, ka, . . . qa * bib, jb, kb, . . . qb + aia, ja, ka, . . . qa * cic, jc, kc, Therefore, M2 = multidimensional matrix of values m2im2, jm2, . . . qc. km2, . . . qm2 = (bib, jb, kb, . . . qb + cic, jc, kc, . . . qc) * aia, ja, ka, . . . qa where the Therefore, M2 = multidimensional matrix of values m2im2, jm2, indices of each element m2im2, jm2, km2, . . . qm2 are determined by km2, . . . qm2 = aia, ja, ka, . . . qa * bib, jb, kb, . . . qb + aia, ja, ka, . . . qa * cic, jc, kc, a concatenation of the indices for bib, jb, kb, . . . qb or cic, jc, kc, . . . qc . . . qc where the indices of each element m2im2, jm2, km2, . . . qm2 are followed by the indices for aia, ja, ka, . . . qa. determined by a concatenation of the indices for aia, ja, ka, . . . qa Because aia, ja, ka, . . . qa, bib, jb, kb, . . . qb, and cic, jc, kc, . . . qc are scalar followed by the indices for bib, jb, kb, . . . qb or are determined by a values, m2im2, jm2, km2, . . . qm2 = (bib, jb, kb, . . . qb + cic, jc, kc, . . . qc) * aia, concatenation of the indices for aia, ja, ka, . . . qa followed by the ja, ka, . . . qa = bib, jb, kb, . . . qb * aia, ja, ka, . . . qa + cic, jc, kc, . . . qc * aia, ja, ka, indices for cic, jc, kc, . . . qc. . . . qa.
ISBN: 978-988-18210-8-9 WCE 2010 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering 2010 Vol III WCE 2010, June 30 - July 2, 2010, London, U.K.
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 or are 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 indices of multidimensional matrix N2 are in2, jn2, kn2, . . . qn2. The indices of 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, . . . qb followed 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 or are 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 Raton, Fla.: CRC.
ISBN: 978-988-18210-8-9 WCE 2010 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)