Page 58 - 35Linear Algebra

58 Systems of Linear Equations

The inverse of the ERO matrices (corresponding to the description of the reverse row
maniplulations)

     
0 1 0 2 0 0 1 0 0
E −1 =  1 0 0  , E −1 =  0 1 0  , E 3 −1 =  0 1 1  .
1
2
0 0 1 0 0 1 0 0 1
Multiplying these gives

     
0 1 0 2 0 0 1 0 0
E −1 E −1 E −1 =  1 0 0   0 1 0   0 1 1 
1 2 3
0 0 1 0 0 1 0 0 1
     
0 1 0 2 0 0 0 1 1
=  1 0 0   0 1 1  =  2 0 0  = M .
0 0 1 0 0 1 0 0 1

2.3.4 LU, LDU, and PLDU Factorizations

The process of elimination can be stopped halfway to obtain decompositions
frequently used in large computations in sciences and engineering. The first
half of the elimination process is to eliminate entries below the diagonal
leaving a matrix which is called upper triangular. The elementary matrices
which perform this part of the elimination are lower triangular, as are their
inverses. But putting together the upper triangular and lower triangular
parts one obtains the so-called LU factorization.

Example 29 (LU factorization)

   
2 0 −3 1 2 0 −3 1
0 1 2 2 0 1 2 2
  E 1  
M =   ∼  
 −4 0 9 2   0 0 3 4 
0 −1 1 −1 0 −1 1 −1
   
2 0 −3 1 2 0 −3 1
0 1 2 2   0 1 2 2 

E 2 E 3
∼   ∼   := U ,
0 0 3 4 0 0 3 4
   
0 0 3 1 0 0 0 −3
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