Matrix – Definition and Basic Operations

Motivation:

 

         Consider the following system equation

                           

 

         Is it possible to have concise notation to describe the system equation ?

         What if we represent it as follows:

 

                                =       

         

         To be specific, if we define

                        A =      v =    d =   ,

 

          Then the following representation is more concise :

 

                                   Av = d

 

        It looks like “something” is operating on     and the result of the operation is     .    And the original problem of the system equation turns out to be finding .

                      

 

              What is the benefit to have notation like this?   The following scenario is used to give a flavor of its advantage :

 

                 and         ,  find    

 

           It is not fun to find the solution to x and y and use the information to solve another system equation.   If we represent the question in the following way

 

              =    and   =

 

 

            It could be written as follows :

 

                                =    

 

            Eventually, we would like to have expression like

                             

                                =   

 

            The reason why we would like to have form like this is that we can map it to the old style to write a system equation.  Thus, the rest of work is how we define a good framework such that the framework could allow us to figure out how to properly handle the expression

 

                         =    

 

          Matrix theory will be provided as a good tool to resolve those issues.

 

 

 

 Definition ( Matrix )

                 Matrix is a two dimensional array with finite number of rows and columns.   We denote the element at ith row and jth column as  a_{ij .}  The whole matrix can be written as {a_ij}  or  

                                     

                                           if it has m rows and n columns.

The number of rows does not have to be equal to the number of columns and we say that it is an m-by-n ( denoted as m  n ) matrix .

 

 

Definition ( Matrix Addition and Subtraction )

 

                  Given two matrices  A ( { a_ij } ) and  B ( {b_ij} ) ,  they  can be added or subtracted only when they are of the same number of rows and the same number of columns.   The addition is defined by

 

                                      A + B = { c_ij }, 

             

               where c_ij = a_ij + b_ij  for the element at ith row and jth column.

 

               For example,

 

                             +   =   = 

 

              Subtraction is defined by the similar way.

 

Definition (  Matrix multiplied by scalar )

               Let  A ( {a_ij} ) be a matrix.  What is  A+A ?  We would like to write it as  2A such that each element in the matrix  2A  is two times the corresponding element in the matrix A.  So, for each scalar c, 

 

                  cA  =  {ca_ij}

 

             For example,  if

                             A =   

             then

                           6A = 

 

 

 

 

 

Definition ( Matrix Multiplication )

 

                 Assume we have two matrices  A ( { a_ij } ) and  B ( {b_ij} )  such that A is an  m  n matrix and B is a p  q matrix.   If   n  p ,  the multiplication of the two matrices A and B does not exist.  The multiplication is only defined when n = p.

 

                If  n = p, we define the multiplication of the two matrices A and B as

 

                                   C = AB = {c_ij}

 

                    Where 

                                    c_ij = 

 

           For example,  if

                         A=  ,  B =  

 

               then

                          AB =    =   = 

 

 

          Please notice : in Matrix operation,  AB is not necessary equal to BA.  In our example here,  BA does not even exist.

 

 

Example :

                       =