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• Image Processing: Digital images can be represented using matrices. In the following figure, each box
                   is and appears transparent or represented in the corresponding matrix with either a “1” or a “0”. A zero
                   means the box contains nothing white, while a one indicates that the box is filled (in this case it is filled
                   with black).


                                                                           0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
                                                                           0  1  1  1  0  0  0  0  0  0  0  0  0  1  1  1  0
                                                                           0  1  1  1  1  0  0  0  0  0  0  0  1  1  1  1  0
                                                                           0  1  1  0  1  1  0  0  0  0  0  1  1  0  1  1  0
                                                                           0  1  1  0  0  1  1  0  0  0  1  1  0  0  1  1  0
                                                                           0  1  1  0  0  0  1  1  1  1  1  0  0  0  1  1  0
                                                                           0  1  1  0  0  0  0  1  1  1  0  0  0  0  1  1  0
                                                                           0  1  1  0  0  0  0  0  0  0  0  0  0  0  1  1  0
                                                                           0  1  1  0  0  0  0  0  0  0  0  0  0  0  1  1  0
                                                                           0  1  1  0  0  0  0  0  0  0  0  0  0  0  1  1  0
                                                                           0  1  1  0  0  0  0  0  0  0  0  0  0  0  1  1  0
                                                                           0  1  1  0  0  0  0  0  0  0  0  0  0  0  1  1  0
                                                                           0  1  1  0  0  0  0  0  0  0  0  0  0  0  1  1  0
                                                                           0  1  1  0  0  0  0  0  0  0  0  0  0  0  1  1  0
                                                                           0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0


                    • Recommender Systems: They use matrices to relate between users and the purchased or viewed item(s).

                    • Natural Language Processing: In NLP, vectors represent the distribution of a particular word in a document.
                   Vectors are one-dimensional matrices.


                        Operations on Matrices


                 Addition of Matrices
                 The sum of two matrices is a matrix obtained by adding the corresponding elements of the given matrices. Also, the
                 two matrices have to be of the same order. For example:
                                                            2   4        1  0
                                                                         
                                                       A =      6  8     B = − 2  7   
                                                                         
                                                             7   5     32      3  4     32
                                                                  ×
                                                                                 ×
                                                              21   4 0   +    3  4 
                                                              +
                                                                               
                                                        +
                                                      A B =   6+−   8 7 ⇒    4  15 
                                                                      +
                                                                         
                                                                ( ) 2
                                                                            10 9 
                                                                      +
                                                              +
                                                              7 3   54         
                 Difference of Matrices
                 If A and B are two matrices of the same order, then the difference A – B is defined as a matrix where each element is
                 obtained by subtracting the corresponding elements (a  – b ). For example:
                                                                  ij  ij
                                                           12 3        3  2  3−
                                                      A =         B =         
                                                           230    23    −  1 0 −  2   23
                                                                                  ×
                                                                  ×
                                                      13   2 −−     3 3        2 40−
                                                                       −
                                                      −
                                                               ( ) 2
                                               AB   −=                       ⇒       
                                                                        ( ) 2 
                                                     − −    −       0 − −      3  32   23
                                                     2 ( 1) 3 0
                                                    
                                                                                          ×
                 Transpose of a Matrix
                                                                                                                        T
                 A matrix obtained by interchanging the rows and columns of a matrix. Transpose of a matrix A is denoted by A’ or A .
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