Proceedings of the World Congress on 2010 Vol III WCE 2010, June 30 - July 2, 2010, London, U.K.

Multidimensional : Multidimensional Matrix Transpose, Symmetry, Antisymmetry, , and Inverse, Part 4 of 6

Ashu M. G. Solo

 Let x and y be any two 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   can be represented as multidimensional matrices or vice        a211111 a 221111     a 211112 a 221112   versa. Some Cartesian and general 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, , 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 and second dimension of multidimensional matrix A, the resulting 6-D A. Multidimensional Matrix Transpose 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   T(x, y) , matrix A is indicated by A . a a   a a  112111 212111     112112 212112   A multidimensional matrix transpose can be done for any     a122111 a 222111     a 122112 a 222112   two dimensions. For each element in the initial  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, matrix E has dimensions of 1 * 2 * 2 * 2 * 2 * 2. If E = AT(1, 4), element of A and for any dimensions x and y, then (A + B) y) = AT(x, y) + BT(x, y). 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.

1 2   4 5  The multidimensional matrix determinant of the following  ,   5-D matrix with dimensions of 2 * 2 * 2 * 2 * 2 results in a 3-D 2 3   5 6  B =  matrix with dimensions of 2 * 2 * 2 and is calculated as 7 8   10 11  follows:  ,   8 9   11 12  1 5   6 2   ,   The following multidimensional matrix C is symmetric 8 0   4 5  with respect to the two indices j and k because cijkl = cikjl for all  2 3   0 7  1 0  5  8 6  5  2  4 i, j, k, and l: , 7 6   9 4   1 2   5 6      2 6  3  7 0  4  7  9 , =     9 2 8 0 9 1  2  8 8  2  0  5 3 4   7 8       C =  ,    2 9   6 11  8 1   5 2  6 0  7  3 1*2 - 6*3  ,    4 10   8 12  6 7   1 6    ,   The following multidimensional matrix D is antisymmetric 3 0   3 2  with respect to the two indices i and j because d = -d for ijkl jikl 40 22 all i, j, k, and l:  9 63 0 1   0 2  =   ,   7 16 1 0   2 0   D = 21 16 0 3   0 4   , The multidimensional matrix determinant of the following 3 0   4 0      3-D matrix with dimensions of 2 * 4 * 2 results in a 1-D matrix For multidimensional matrix D to be antisymmetric (dijkl = with two undefined elements: -d ), it is necessary that d = 0 for all i, k, and l. That is, d jikl iikl 11kl 1 6 4 7 = 0, d22kl = 0 for all k and l.  The following multidimensional matrix E is antisymmetric 4 0 9 6 undefined =  with respect to the two indices j and k because eijkl = -eikjl for 5 3 1 7 undefined all i, j, k, and l:  3 2 2 8 0 1   0 3   ,   A multidimensional matrix determinant is simply a 0 2   0 4  multidimensional matrix of for each 2-D square E =  1 0   3 0  submatrix within the multidimensional matrix. , The properties of determinants in classical matrix algebra 2 0   4 0      apply to each 2-D square submatrix within the For multidimensional matrix E to be antisymmetric (eijkl = multidimensional matrix: T -eikjl), it is necessary that eijjl = 0 for all i, j, and l. That is, ei11l For a 2-D square submatrix A, |A| = |A | = 0, ei22l = 0 for all i and l. If all the elements of any row or column are zero in a 2-D square submatrix A, then |A| = 0. If one row is proportional to another row of a 2-D square IV. MULTIDIMENSIONAL MATRIX DETERMINANT submatrix A, then |A| = 0. The multidimensional matrix determinant for a 1-D matrix If one column is proportional to another column of a 2-D is undefined. The multidimensional matrix determinant of a square submatrix A, then |A| = 0. 2-D results in a and is calculated using If one row is a of one or more other rows the standard means in classical matrix algebra. The of a 2-D square submatrix A, then |A| = 0. multidimensional matrix determinant of a 2-D nonsquare If one column is a linear combination of one or more other matrix is undefined. For the multidimensional matrix columns of a 2-D square submatrix A, then |A| = 0. determinant of multidimensional matrices with three or more If two rows of a 2-D square submatrix A are interchanged, the dimensions, each 2-D square submatrix is replaced by its sign of the determinant of submatrix A is changed. scalar determinant and each 2-D nonsquare submatrix is If two columns of a 2-D square submatrix A are interchanged, replaced by an undefined element. the sign of the determinant of submatrix A is changed. Consider two multidimensional matrices A and B where B If all elements of a row of a 2-D square submatrix A are is the multidimensional matrix determinant of A: multiplied by a scalar , the determinant of submatrix A is BA multiplied by |. Multidimensional matrix determinant B has two If all elements of a column of a 2-D square submatrix A are dimensions less than multidimensional matrix A: multiplied by a scalar , the determinant of submatrix A is q(B) = q(A) – 2 for q(A)  3 where q represents the number multiplied by  of dimensions of the matrix in parentheses. Any multiple of a row of a 2-D square submatrix A can be added to any other row without changing the value of the Nd(B) = Nd+2(A) for d+2 = 3, . . ., q(A) where N represents the number of elements in the subscripted dimension for the determinant for submatrix A. matrix in parentheses and q(A) is the number of dimensions of Any multiple of a column of a 2-D square submatrix A can be multidimensional matrix A. added to any column without changing the value of the determinant for submatrix A.

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.

matrix has an inverse and that multidimensional matrix is multiplied by its inverse in the first dimension and second V. MULTIDIMENSIONAL MATRIX INVERSE dimension, then the product will be a multidimensional -1 identity matrix with the same dimensions. That is, A *(1, 2) A The multidimensional matrix inverse for a 1-D matrix is -1 -1 -1 undefined. The multidimensional matrix inverse of a 2-D = A *(1, 2) A = AA = A A = UNIT. Multiplying the first dimension and second dimension of matrix exists if it is a square matrix and has a nonzero -1 determinant, and is calculated using the standard means in multidimensional matrix A by multidimensional matrix A classical matrix algebra. For the multidimensional matrix results in a multidimensional identity matrix UNIT(2 * 2 * 2 * 2, 1, inverse of multidimensional matrices with three or more 2): dimensions, each 2-D square submatrix with a nonzero determinant is replaced by its inverse and any other 2-D submatrices or 1-D submatrices are replaced by an undefined element. Consider the following 4-D matrix A with dimensions of 2 AA-1 = * 2 * 2 * 2: 1 3   9 11   ,   2 4   10 12  A =  5 7   13 15   ,   1 0   1 0  6 8   14 16   ,   0 1   0 1  The four 2-D square submatrices of A are = = UNIT(2 * 2 * 2 * 2, 1, 2) 1 0   1 0  13 57 9 11  ,   A11 = ; A21 = ; A12 = ; 0 1   0 1  24 68 10 12

13 15 A22 =  14 16 VI. CONCLUSION The inverse of each 2-D square submatrix is calculated Part 4 of 6 defined the multidimensional matrix algebra using standard methods in classical matrix algebra: operations for transpose, determinant, and inverse. Also, part 1 T 4 of 6 defined multidimensional matrix symmetry and 13 42 -1 Adj(A11 )  1  antisymmetry. A11 = =  =  A11 24 2 31 Part 5 of 6 describes the commutative, associative, and distributive laws of multidimensional matrix algebra. 3 2 2 =  1 1 REFERENCES  2 [1] Franklin, Joel L. [2000] Matrix Theory. Mineola, N.Y.: Dover. The inverses of the other three 2-D square submatrices in A [2] Young, Eutiquio C. [1992] Vector and Tensor Analysis. 2d ed. Boca are as follows: Raton, Fla.: CRC.

7 11 15 4 6 8 -1 2 -1 2 -1 2 A21 = ; A12 = ; A22 =  5 9 13 3 5 7 2 2 2 The inverse of multidimensional matrix A consists of the four inverted 2-D square submatrices: 3   11  26 22     ,   19 15    22    A-1 =  7   15  48 22     ,   5 13 37    22    In classical matrix algebra, if a matrix has an inverse and that matrix is multiplied by its inverse, the product is an identity matrix with the same dimensions. That is, AA-1 = A-1A = I. The same applies to each 2-D submatrix in a multidimensional matrix. Because multidimensional matrices are a concatenation of 2-D submatrices, if a multidimensional

ISBN: 978-988-18210-8-9 WCE 2010 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)