LECTURE8:
           BASICMATRIXALGEBRA
                      Prof. N. Harnew
                    University of Oxford
                          MT2012
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              Outline: 8. BASIC MATRIX ALGEBRA
       8.1 Addition of matrices
          8.1.1 Properties and an example
       8.2 Multiplication by a scalar
          8.2.1 Example
       8.3 Multiplication of matrices
          8.3.1 Example 1
          8.3.2 Example 2
       8.4 Properties of matrix multiplication
          8.4.1 Proof of A(BC) = (AB)C
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                                                         8.1 Addition of matrices
                    ◮ Matrix summation follows the properties of linear algebra
                                                                                      C=A+B
                                                            Hence C =(A+B) =A +B
                                                                                    ij                        ij          ij          ij
                    ◮ Writing this out:
                                                      A                A           · · ·      A          B                       B           · · ·      B         
                                                              11           12                     1n                       11          12                     1n
                                                      A                A           · · ·      A          B                       B           · · ·      B         
                            C=A+B= 21                                     22                     2n + 21                            22                     2n 
                                                      ···              · · ·       · · ·       · · ·     ···                      · · ·      · · ·       · · ·    
                                                           A            A           · · ·      A                        B           B           · · ·      B
                                                              m1          m2                     mn                       m1           m2                     mn
                                                                                                                                                                   (1)
                                                        A11+B11                       A12 +B12               · · ·       A1n +B1n 
                                                  = A21+B21                           A22 +B21               · · ·       A2n +B2n                                  (2)
                                                                   · · ·                    · · ·            · · ·             · · ·          
                                                             Am1 +Bm1                 Am2 +Bm1                · · ·      Amn +Bmn
                    ◮ For this to have any meaning, both matrices must have the same
                          dimensions (in this case m × n).
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                    8.1.1 Properties and an example
         ◮ It follows obviously (and also from the rules of linear
            operators) that
            C=A+B=B+A(commutative).
         ◮ The difference of two matrices follows in an obvious way:
            C=A−BhaselementsC =A −B .
                                        ij    ij    ij
       Example
       C=A+B= 1 2 3 + 7                    8   9 = 8 10 12 
                       4 5 6           −2 −1 0              2   4    6(3)
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