Proceedings of the World Congress on Engineering 2010 Vol III WCE 2010, June 30 - July 2, 2010, London, U.K. Multidimensional Matrix Mathematics: Multidimensional Matrix Transpose, Symmetry, Antisymmetry, Determinant, and Inverse, Part 4 of 6 Ashu M. G. Solo Let x and y be any two dimensions being transposed. If B = T(x, y) Abstract—This is the first series of research papers to define A , then bijk . q with x & y transposed = aijk . q. T(1, 5) multidimensional matrix mathematics, which includes For example, if B = A , then bmjklin . q = aijklmn . q. multidimensional matrix algebra and multidimensional matrix Consider the 6-D matrix A with dimensions of 2 * 2 * 2 * 1 calculus. These are new branches of math created by the author * 2 * 2: with numerous applications in engineering, math, natural science, social science, and other fields. Cartesian and general a111111 a 121111 a 111112 a 121112 tensors can be represented as multidimensional matrices or vice a211111 a 221111 a 211112 a 221112 versa. Some Cartesian and general tensor operations can be , a a a a performed as multidimensional matrix operations or vice versa. 112111 122111 112112 122112 However, many aspects of multidimensional matrix math and a212111 a 222111 a 212112 a 222112 tensor analysis are not interchangeable. Part 4 of 6 defines the A = multidimensional matrix algebra operations for transpose, aa111121 121121 aa111122 121122 determinant, and inverse. Also, multidimensional matrix aa211121 221121 aa211122 221122 symmetry and antisymmetry are defined. , a112121a 122121 a 112122 a 122122 Index Terms—multidimensional matrix math, a a a a multidimensional matrix algebra, multidimensional matrix 212121 222121 212122 222122 calculus, matrix math, matrix algebra, matrix calculus, tensor 1 2 9 10 analysis 3 4 11 12 , 5 6 13 14 I. INTRODUCTION 7 8 15 16 Part 4 of 6 defines the multidimensional matrix algebra = operations for transpose, determinant, and inverse. Also, part 17 18 25 26 4 of 6 defines multidimensional matrix symmetry and 19 20 27 28 antisymmetry. , 21 22 29 30 23 24 31 32 II. MULTIDIMENSIONAL MATRIX TRANSPOSE If we take the transpose of the first dimension and second dimension of multidimensional matrix A, the resulting 6-D A. Multidimensional Matrix Transpose Operation matrix C has dimensions of 2 * 2 * 2 * 1 * 2 * 2. If C = AT(1, 2) A multidimensional matrix transpose is indicated by the , then cjiklmn = aijklmn. letter “T” followed by the dimensions being transposed in C = AT(1, 2) = parentheses, all of which is a superscript to the a111111 a 211111 a 111112 a 211112 multidimensional matrix being transposed. The transpose of the xth dimension and yth dimension of multidimensional a121111 a 221111 a 121112 a 221112 , matrix A is indicated by AT(x, y). a112111 a 212111 a 112112 a 212112 A multidimensional matrix transpose can be done for any a a a a two dimensions. For each element in the initial 122111 222111 122112 222112 multidimensional matrix, the two dimensions are transposed aa111121 211121 aa111122 211122 and the element is placed at the resulting location in the aa121121 221121 aa121122 221122 resulting multidimensional matrix. , a a a a 112121 212121 112122 212122 Manuscript received March 23, 2010. a122121 a 222121 a 122122 a 222122 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]). 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. 1 3 9 11 C. Proofs of Multidimensional Matrix Transpose Laws The following definitions are used in the following 2 4 10 12 , transpose laws and their proofs: 5 7 13 15 Eijk . q(A) refers to the element with indices i, j, k, . ., q in 6 8 14 16 multidimensional matrix A. = E (AT(x, y)) refers to the element with ijk . q with x & y transposed 17 19 25 27 indices i, j, k, . ., q, but with the xth dimension and yth 18 20 26 28 dimension transposed. The xth dimension and yth dimension , 21 23 29 31 can be any two different dimensions. For example, if x=2 and y=4, then the second and fourth 22 24 30 32 dimensions of multidimensional matrix A are being In classical matrix algebra, the transpose operation always transposed and for any element of A, Eijkl . q with x & y T(x, y) T(2, 4) involves the first dimension and second dimension because transposed(A ) = Eilkj . q(A ) = Eijkl . q(A). there are no other dimensions involved in classical matrix Multidimensional matrix transpose law #1, (AT(x, y))T(x, y) = algebra. In multidimensional matrix algebra, when the first A, is proven as follows: dimension and second dimension of multidimensional For any element of A and for any dimensions x and y: T(x, y) T(x, y) T(x, y) matrices are being transposed, the subscripted dimensions in Eijk . q[(A ) ] = Eijk . q with x & y transposed(A ) parentheses “(1, 2)” by the multiplication sign can be omitted. = Eijk . q(A) T(1, 2) T T(x, y) T(x, y) That is, A = A . This makes the notation for transpose of Because Eijk . q[(A ) ] = Eijk . q(A) for any element of 2-D matrices in multidimensional matrix algebra consistent A and for any dimensions x and y, then (AT(x, y))T(x, y) = A. with classical matrix algebra. Multidimensional matrix transpose law #2, (x, y) If we take the transpose of the third dimension and fourth (x, y) is proven as follows: dimension of multidimensional matrix A, the resulting 6-D For any element of A and for any dimensions x and y: T(3, (x, y) matrix D has dimensions of 2 * 2 * 1 * 2 * 2 * 2. If D = A Eijk . q[ ] = Eijk . q with x & y transposed 4) , then dijlkmn = aijklmn. = E T(3, 4) ijk . q with x & y transposed D = A = (x, y) Eijk . q a111111 a 121111 a 112111 a 122111 a 111112 a 121112 a 112112 a 122112 (x, y) (x, y) ,,, Because Eijk . q[ ] = Eijk . q for any a211111 a 221111 a 212111 a 222111 a 211112 a 221112 a 212112 a 222112 (x, element of A and for any dimensions x and y, then ( a111121 a 121121 a 112121 a 122121 a 111122 a 121122 a 112122 a 122122 y) (x, y) , ,, aa211121 221121 a212121 a 222121 a 211122 a 221122 a 212122 a 222122 Multidimensional matrix transpose law #3, (A + B)T(x, y) = 12 56 910 1314 AT(x, y) + BT(x, y), is proven as follows: ,,, 3 4 7 8 11 12 15 16 For any element of A and for any dimensions x and y: T(x, y) = Eijk . q[(A + B) ] = Eijk . q with x & y transposed(A + B) 17 18 21 22 25 26 29 30 ,,, = Eijk . q with x & y transposed(A) + Eijk . q with x & y transposed(B) T(x, y) T(x, y) 19 20 23 24 27 28 31 32 = Eijk . q(A ) + Eijk . q(B ) T(x, y) T(x, y) = Eijk . q(A + B ) If we take the transpose of the first dimension and fourth T(x, y) T(x, y) T(x, y) Because Eij[(A + B) ] = Eij(A + B ) for any dimension of multidimensional matrix A, the resulting 6-D T(x, T(1, 4) element of A and for any dimensions x and y, then (A + B) matrix E has dimensions of 1 * 2 * 2 * 2 * 2 * 2. If E = A , y) T(x, y) T(x, y) = A + B . then eljkimn = aijklmn. E = AT(1, 4) = a111111 a 121111,, a 211111 a 221111 a 111112 a 121112 a 211112 a 221112 , III. SYMMETRY AND ANTISYMMETRY OF a112111 a 122111,, a 212111 a 222111 a 112112 a 122112 a 212112 a 222112 MULTIDIMENSIONAL MATRICES a111121 a 121121, a 211121 a 221121 a111122 a 121122, a 211122 a 221122 , Just like a tensor can be symmetric or antisymmetric with a112121 a 122121, a 212121 a 222121 a112122 a 122122, a 212122 a 222122 respect to any two indices, a multidimensional matrix can be 12,34 910,1112 symmetric or antisymmetric with respect to any two indices. , Let da and db be any two dimensions being transposed. A 5 6,7 8 1314,1516 = multidimensional matrix A is symmetric with respect to the 17 18 , 19 20 25 26 , 27 28 two indices da and db if aijklmnop = aijklmnop with da & db transposed. A , multidimensional matrix A is antisymmetric with respect to 21 22 , 23 24 29 30 , 31 32 the two indices da and db if B. Transpose Laws of Multidimensional Matrix Algebra aijklmnop = - aijklmnop with da & db transposed. For example, a multidimensional matrix A is symmetric The multidimensional matrix transpose laws follow: with respect to the two indices i and j if a = a . Multidimensional Matrix Transpose Law #1: ijklmnop jiklmnop T(x, y) T(x, y) Also, for example, a multidimensional matrix A is (A ) = A antisymmetric with respect to the two indices i and j if a Multidimensional Matrix Transpose Law #2: ijklmnop (x, y) (x, y) = -ajiklmnop. The following multidimensional matrix B is symmetric Multidimensional Matrix Transpose Law #3: T(x, y) T(x, y) T(x, y) with respect to the two indices i and j because bijkl = bjikl for all (A + B) = A + B i, j, k, and l: 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.