Matrix – Row Operation, Column Operation, and Some Basic Properties

 

Motivation:

                       

 

                          The matrix multiplication has been shown in the previous section. In this section, we are going to find out more properties on it.  Consider the following matrix operation:

 

                                  P = 

 

                   It is the multiplication of  1  3 matrix and 3  3 matrix. So,  the result is a 1  3 matrix. Instead of finding each entry directly, it is equivalent to the following way:

 

                         P= 1  + 2  + 3

 

                   How about the following multiplication:

 

                                                Q=

 

                   With similar approach, we can write it as

 

                                              Q= 1  +2  +3

 

                   With these two basic principles in mind, we would like to ask one question: what if we would to multiply the first row by 1,  add the second row, and put the result to replace the original second row ?  Is there any “corresponding matrix” to do the operations mentioned above?  We are going to find out.

 

 

Identity Matrix

                  For any n  n matrix P,  does exist any matrix Q such that  PQ = P  and QP = P? The answer is yes.  With all the diagonal entries equal to 1 and the rest equal to 0,  this matrix can satisfy what we are looking for:

 

                                                 

                But is it possible that there exists other matrix that also satisfies the condition we are looking for?   This problem involves that the rows or columns of P are “linearly independent” or not – if you recall the two tricks at the beginning of this section.  We define “linearly independent” below.

 

 

Definition ( Linearly Independent )

                For a set of vectors  v₁, v₂, v₃, … , v_n ,  when the following condition is satisfied, we say that  v₁, v₂, v₃, … , v_n  are linearly independent:

 

               c₁v₁ + c₂v₂ + c₃v₃ + … + c_nv_n = 0, where c₁, c₂, ..., c_n are scalars

         if and only if    c₁=0, c₂=0, … , c_n = 0

 

               Otherwise, they are called “linearly dependent” .

 

Example 1:   The two vectors (1,0) and (0,1) are linearly independent.

 

Example 2: (1,2) and (2,4) are linearly dependent.

 

          Basically,  “linearly independent” means a vector in that set can not be represented by linear combination of the other vectors.

 

 

 

Definition ( Identity Matrix )

               For an n  n matrix with all the diagonal elements equal to 1 and the rest of the elements equal to 0,  we call it an “identity matrix”.  And we denote it as I_n .

 

 

 

 

Elementary Row Operation

 

                   Consider the following matrix operation.

 

 

 

                     Let

                                         U =

       

 

 

 

 

                  So,          

                                 U

                                =

                              = (  +  )

                              =  +

 

                         So, multiplying U from the right is equivalent to put the sum of the first row and second row into the position of second row.  In general, we have the following theorem.

 

Theorem ( Row Operation Matrix )

                 For an Identity Matrix I_n ,  if an operation on this identity matrix I_n is performed by  multiplying row j by a constant b and adding row i, and put the sum on row i to form a new matrix U.   Then for an n  n matrix A,  UA  is the result that you perform the same operation on the matrix A.

 

 

 

 

Elementary Column Operation

                 Similarly, let’s check the following operation:

 

                               R =

                 Thus,

                                     R

                          =  (  +  )

                          =  +

 

                     Via this observation, we know that multiplying matrix R from the right is equivalent to perform the corresponding column operation.

 

 

Theorem ( Column Operation Matrix )

                For an identity matrix I_n ,  if we perform an operation on this identity matrix by multiplying column j and then adding column i, and then put the result into column i to form a new matrix R.  Then for an n  n matrix A,  AR is the result that you perform the same column operation on matrix A.