Matrices, transposes, and inverses
                    Math 40, Introduction to Linear Algebra
                    Wednesday, February 1, 2012
                    Matrix-vector multiplication:  two views
                     • 1st perspective:  A   is linear combination of columns of A
                                              ￿x
                                            
                         ￿               ￿           ￿ ￿         ￿    ￿       ￿ ￿
                         ￿              ￿ 4            ￿ ￿        ￿     ￿       ￿ ￿       ￿ ￿
                           1    −2 3 4                   1          −2            3
                          1    −23                       1          −2           3           4
                                            
                                           
                                             3 =4            +3            +2
                           215 3 =4 2 +3 1 +2 5 =
                          2152 2                                     1           5          21
                                A            2
                                             ￿x
                     • 2nd perspective:  A   is computed as dot product of rows of A with vector
                                               ￿x                                                                 ￿x
                         ￿               ￿4                        4                     ￿ ￿
                           1    −23                                   ￿ ￿       ￿ ￿           4
                           2     1     5 3=                          2          4  =       21
                                             2         dot product of   1   and    3
                                A                                       5          2
                                             ￿x
                       Notice that # of columns of A = # of rows of    .
                                                                                    ￿x
                       This is a requirement in order for matrix multiplication to be defined.
           Matrix multiplication 
            What sizes of matrices can be multiplied together? 
            For m x n matrix A and n x p matrix B,  the matrix product AB 
                                               is an m x p matrix.
                               m x n   n x p
                                   “inner” 
                                  parameters 
                                  must match
                             “outer” parameters become 
                              parameters of matrix AB
            If A is a square matrix and k is a positive integer, we define
                               Ak = A·A···A
                                   ￿   ￿￿  ￿
                                     k factors
           Properties of matrix multiplication 
           Most of the properties that we expect to hold for matrix multiplication do....
                            A(B+C)=AB+AC
                              (AB)C =A(BC)
                       k(AB)=(kA)B =A(kB) for scalar k
                           .... except commutativity!!
                          In general, AB ￿= BA.
               Matrix multiplication not commutative
               Problems with hoping AB and BA are equal:             In general, 
                  • BA may not be well-defined.                       AB￿=BA.
                       (e.g., A is 2 x 3 matrix, B is 3 x 5 matrix)
                  • Even if AB and BA are both defined, AB and BA  may not be 
                    the same size.
                       (e.g., A is 2 x 3 matrix, B is 3 x 2 matrix)
                  • Even if AB and BA are both defined and of the same size, they 
                    still may not be equal.
                       ￿     ￿￿     ￿    ￿    ￿   ￿     ￿   ￿     ￿￿      ￿
                        1112 24 33 1211
                                      =         ￿=        =
                        1112 24 33 1211
               Truth or fiction?
               Question 1    For n x n matrices A and B, is 
                                    2     2
                                  A −B =(A−B)(A+B)?
                             (A−B)(A+B)=A2+AB−BA−B2
                  No!!                                AB−BA
                                                      ￿  ￿￿   ￿
                                                         ￿=0
                                                                 2    2  2
               Question 2    For n x n matrices A and B, is (AB) = A B ?
                                     2                         2  2
                  No!!         (AB) =ABAB￿=AABB=A B
                        Matrix transpose
                                           The transpose of an m x n matrix A is the n x m matrix 
                         Definition
                          AT obtained by interchanging rows and columns of A,
                                                                       T
                                                           i.e., (A )          =A ∀i,j.
                                                                           ij         ji
                         Example                                                                                
                                          ￿135−2￿                                                       15
                                                                                                                
                                  A=                                                         T          33
                                            5321                                          A =                   
                                                                                                                
                                                                                                        52
                                Transpose operation can be viewed as                                   −21
                                  flipping entries about the diagonal.
                         Definition                 A square matrix A is symmetric if AT = A.
                        Properties of transpose
                                                                                               apply twice -- get back
                                  (1)             T T                                           to where you started
                                            (A ) =A
                                  (2)                         T              T            T
                                            (A+B) =A +B
                                  (3) For a scalar c,(cA)T = cAT
                                  (4)                   T              T T
                                            (AB)              =B A
                                                                                               To prove this, we show that
                                                                                                         T
                                                                                                 [(AB) ] =
                        Exercise                                                                            ij   .
                                                                                                                 .
                          Prove that for any matrix A, ATA is symmetric.                                         .
                                                                                                                =[(BTAT)]
                                                                                                                                 ij