Page 61 - 35Linear Algebra

2.4 Review Problems 61

The LDU factorization of a matrix is a factorization into blocks of EROs
of a various types: L is the product of the inverses of EROs which eliminate
below the diagonal by row addition, D the product of inverses of EROs which
set the diagonal elements to 1 by row multiplication, and U is the product
of inverses of EROs which eliminate above the diagonal by row addition.
You may notice that one of the three kinds of row operation is missing
from this story. Row exchange may be necessary to obtain RREF. Indeed, so
far in this chapter we have been working under the tacit assumption that M
can be brought to the identity by just row multiplication and row addition.
If row exchange is necessary, the resulting factorization is LDPU where P is
the product of inverses of EROs that perform row exchange.

Example 31 (LDPU factorization, building from previous examples)

   
0 1 2 2 2 0 −3 1
 2 0 −3 1  P  0 1 2 2 
∼
M =   ∼   E 6 E 5 E 4 E 3 E 2 E 1 L
−4 0 9 2 −4 0 9 2
   
0 −1 1 −1 0 −1 1 −1

 
0 1 0 0
 1 0 0 0  −1
P =   = P
0 0 1 0
 
0 0 0 1

M = P(E −1 E −1 E −1 )(E −1 E −1 E −1 )(E −1 )U = PLDU
1 2 3 4 5 6 7

      3 1 
0 1 2 2 1 0 0 0 2 0 0 0 0 1 0 0 1 0 −
2 2
 2 0 −3 1   0 1 0 0  0 1 0 0  1 0 0 0  0 1 2 2 
  =     
 −4 0 9 2   −2 0 1 0  0 0 3 0  0 0 1 0  0 0 1 4 
3
0 −1 1 −1 0 −1 1 1 0 0 1 −3 0 0 0 1 0 0 0 1
2.4 Review Problems

Reading problems 3
Webwork: Matrix notation 18
LU 19

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