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
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                                                                                                 5 2 8 7
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                                                                                             
                                         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 114 25
                                                                                                
                                                                                             
               x11 + 5x21 + 3x31 = 10                                                                                ~                                
                                                                                             
                                                                                                
               2x  + 7x  + 4x  = 15                                                              10 4 313                  1 0 013112
                  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               
                                                                                                                                                       secxxtan
                                                                                                            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
                         secxxtan
                       
                                                        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)