Page 164 - 35Linear Algebra

164 Matrices

The product of lower triangular matrices is always lower triangular!

Moreover it is obtained by recording minus the constants used for all our
row operations in the appropriate columns (this always works this way).
Moreover, U 2 is upper triangular and M = L 2 U 2 , we are done! Putting this
all together we have

   
  1 0 0 6 18 3
6 18 3
 1   
M =  2 12 1  1 0 0 6 0 .
3
= 
4 15 3 2 1 1 0 0 1
3 2
If the matrix you’re working with has more than three rows, just continue
this process by zeroing out the next column below the diagonal, and repeat
until there’s nothing left to do.

Another LU decomposition example

The fractions in the L matrix are admittedly ugly. For two matrices
LU, we can multiply one entire column of L by a constant λ and divide the
corresponding row of U by the same constant without changing the product
of the two matrices. Then:

   
1 0 0 6 18 3
 1 1 0 I 0 
 
3
LU =  6 0
2 1 1 0 0 1
3 2
     1   
1 0 0 3 0 0 3 0 0 6 18 3
 1   1 
 0 6 0  0  0 6 0 
3 1 0 6 0
= 
2 1 1 0 0 1 0 0 1 0 0 1
3 2
   
3 0 0 2 6 1
=  1 6 0   0 1 0  .
2 3 1 0 0 1
The resulting matrix looks nicer, but isn’t in standard (lower unit triangular
matrix) form.

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