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Proceedings of the World Congress on Engineering 2010 Vol III WCE 2010, June 30 - July 2, 2010, London, U.K. Multidimensional Matrix Mathematics: Solving Systems of Linear Equations and Multidimensional Matrix Calculus, Part 6 of 6 Ashu M. G. Solo 1 5 3 x11 10 Abstract—This is the first series of research papers to define 2 7 4 x21 15 multidimensional matrix mathematics, which includes multidimensional matrix algebra and multidimensional matrix 5 2 8 x31 7 calculus. These are new branches of math created by the author * = 10 4 3 x12 13 with numerous applications in engineering, math, natural science, social science, and other fields. Cartesian and general 8 5 1 x22 18 tensors can be represented as multidimensional matrices or vice 2 6 9 x32 19 versa. Some Cartesian and general tensor operations can be performed as multidimensional matrix operations or vice versa. Any of the different methods used to solve systems of linear However, many aspects of multidimensional matrix math and equations represented with classical matrices can be applied tensor analysis are not interchangeable. Part 6 of 6 describes to each of the submatrices in multidimensional matrices. This the solution of systems of linear equations using includes graphing, the substitution method, the elimination multidimensional matrices. Also, the multidimensional matrix method, Gaussian elimination, Gauss-Jordan elimination, calculus operations for differentiation and integration are Cramer’s rule, LU decomposition, Cholesky decomposition, defined. etc. The preceding system of linear equations can be Index Terms—multidimensional matrix math, represented with the following augmented 3-D matrix: multidimensional matrix algebra, multidimensional matrix calculus, matrix math, matrix algebra, matrix calculus, tensor analysis 1 5 310 2 7 415 5 2 8 7 I. INTRODUCTION 10 4 313 Part 6 of 6 describes the solution of systems of linear 8 5 118 equations using multidimensional matrices. Also, part 6 of 6 defines the multidimensional matrix calculus operations for 2 6 919 differentiation and integration. Using Gauss-Jordan elimination on each of the individual 2-D submatrices in the augmented 3-D matrix, the reduced II. SOLVING SYSTEMS OF LINEAR EQUATIONS WITH row echelon form for this augmented multidimensional matrix MULTIDIMENSIONAL MATRICES can be found: Systems of linear equations can be represented and solved 1 5 310 1 0 0 37 25 2 7 415 0 1 0 51 25 with multidimensional matrices. Consider the following system of linear equations: 5 2 8 7 0 0 114 25 x11 + 5x21 + 3x31 = 10 ~ 2x + 7x + 4x = 15 10 4 313 1 0 013112 11 21 31 8 5 118 0 1 0 318 5x + 2x + 8x = 7 11 21 31 10x + 4x + 3x = 13 12 22 32 2 6 919 0 0 1 25 56 8x + 5x + x = 18 12 22 32 37 51 14 13 2x12 + 6x22 + 9x32 = 19 Therefore, x11 = ; x21 = ; x31 = ; x12 = ; x22 This system of linear equations can be represented with the 25 25 25 112 following multidimensional matrix equation composed of 3-D = 31 ; x = 25 matrices: 8 32 56 Consider the following system of linear equations: a x + a x + a x = b 11111 1111 12111 2111 13111 3111 1111 a x + a x + a x = b Manuscript received March 23, 2010. 21111 1111 22111 2111 23111 3111 2111 Ashu M. G. Solo is with Maverick Technologies America Inc., Suite 808, a x + a x + a x = b 1220 North Market Street, Wilmington, DE 19801 USA (phone: (306) 31111 1111 32111 2111 33111 3111 3111 a x + a x + a x = b 242-0566; email: amgsolo@mavericktechnologies.us). 11121 1121 12121 2121 13121 3121 1121 a x + a x + a x = b 21121 1121 22121 2121 23121 3121 2121 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. a x + a x + a x = b a x + a x + a x = b 31121 1121 32121 2121 33121 3121 3121 11122 1122 12122 2122 13122 3122 1122 a x + a x + a x = b a x + a x + a x = b 11211 1211 12211 2211 13211 3211 1211 21122 1122 22122 2122 23122 3122 2122 a x + a x + a x = b a x + a x + a x = b 21211 1211 22211 2211 23211 3211 2211 31122 1122 32122 2122 33122 3122 3122 a x + a x + a x = b a x + a x + a x = b 31211 1211 32211 2211 33211 3211 3211 11212 1212 12212 2212 13212 3212 1212 a x + a x + a x = b a x + a x + a x = b 11221 1221 12221 2221 13221 3221 1221 21212 1212 22212 2212 23212 3212 2212 a x + a x + a x = b a x + a x + a x = b 21221 1221 22221 2221 23221 3221 2221 31212 1212 32212 2212 33212 3212 3212 a x + a x + a x = b a x + a x + a x = b 31221 1221 32221 2221 33221 3221 3221 11222 1222 12222 2222 13222 3222 1222 a x + a x + a x = b a x + a x + a x = b 11112 1112 12112 2112 13112 3112 1112 21222 1222 22222 2222 23222 3222 2222 a x + a x + a x = b a x + a x + a x = b 21112 1112 22112 2112 23112 3112 2112 31222 1222 32222 2222 33222 3222 3222 a x + a x + a x = b 31112 1112 32112 2112 33112 3112 3112 This system of linear equations can be represented with this single multidimensional matrix equation composed of 5-D matrices: a11111 a12111 a13111 a11121 a12121 a13121 xx1111 1121 bb1111 1121 a21111 a22111 a23111 , a21121 a22121 a23121 xx2111 , 2121 bb2111 , 2121 a31111 a32111 a33111 a31121 a32121 a33121 xx3111 3121 bb3111 3121 a11211 a12211 a13211 a11221 aa12221 13221 xx1211 1221 bb1211 1221 aaa21211 22211 23211 , aaa21221 22221 23221 xx2211 , 2221 bb2211 , 2221 aaa31211 32211 33211 aaa31221 32221 33221 xx3211 3221 bb3211 3221 * = a11112 a12112 a13112 a11122 a12122 a13122 xx1112 1122 bb1112 1122 a21112 a22112 a23112 , a21122 a22122 a23122 xx2112 , 2122 bb2112 , 2122 a31112 a32112 a33112 a31122 a32122 a33122 xx3112 3122 bb3112 3122 a11212 a12212 a13212 a11222 a12222 a13222 xx1212 1222 bb1212 1222 a21212 a22212 a23212 , a21222 a22222 a23222 x2212 , x2222 b2212 , b2222 a31212 a32212 a33212 a31222 a32222 a33222 x3212 x3222 b3212 b3222 When each element aijklm of the coefficient matrix and each multidimensional matrix has the same number of dimensions element bjklm of the product matrix is defined, each element and same number of elements in each dimension as the xjklm of the variable matrix can be calculated. Any of the multidimensional matrix that is integrated. different methods used to solve systems of linear equations A 3-D matrix with dimensions of 2 * 2 * 2 is integrated as represented with classical matrices can be applied to each of follows: the submatrices in multidimensional matrices. The system of 0 sinx K cosx linear equations can be solved like in the preceding example. xy 6x ye 2 xy 3xe dx = x x ey2 e 2xy III. MULTIDIMENSIONAL MATRIX DIFFERENTIATION 2 secxxtan secxtanx sec x In multidimensional matrix calculus, multidimensional matrices are differentiated by finding the derivative of each element in the multidimensional matrix. The result is a multidimensional matrix of derivatives. The resulting V. CONCLUSION multidimensional matrix has the same number of dimensions Part 6 of 6 described the solution of systems of linear and same number of elements in each dimension as the equations using multidimensional matrices. Also, part 6 of 6 multidimensional matrix that is differentiated. defined the multidimensional matrix calculus operations for A 3-D matrix with dimensions of 2 * 2 * 2 is differentiated differentiation and integration. as follows: Classical matrix math offers many benefits not present in tensor analysis for a first or second order tensor, and tensor 0 sinx 5 cosx x analysis for a first or second order tensor offers many benefits 2 x 6xe 3xe d not present in classical matrix math. Similarly, = dx x x multidimensional matrix math offers many benefits not ex2 e 2 present in tensor analysis for tensors of any order, and tensor 2 secxxtan secxtanx sec x analysis for tensors of any order offers many benefits not present in multidimensional matrix math. The author predicts that multidimensional matrix math will IV. MULTIDIMENSIONAL MATRIX INTEGRATION replace classical matrix math in the future when this subject is In multidimensional matrix calculus, multidimensional taught in university courses. The author has made many more matrices are integrated by finding the integral of each element developments in multidimensional matrix math and in the multidimensional matrix. The result is a developed many more innovative applications of multidimensional matrix of integrals. The resulting multidimensional matrix math that will soon be published. 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. Numerous applications have been developed for classical matrix math and its subsets, classical matrix algebra and classical matrix calculus, in many extremely diverse fields. Similarly, numerous applications in many extremely diverse fields will emerge for multidimensional matrix math and its subsets, multidimensional matrix algebra and multidimensional matrix calculus. These new branches of math will make it easier to solve many problems than before and even solve problems that couldn’t be solved before. 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)
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