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.