Page 55 - Matrix Analysis & Applied Linear Algebra

48 Chapter 2 Rectangular Systems and Echelon Forms

Compare the results of this example with the results of Example 2.1.1, and
notice that the “form” of the final matrix is the same in both examples, which
indeed must be the case because of the uniqueness of “form” mentioned in the
previous section. The only difference is in the numerical value of some of the
entries. By the nature of Gauss–Jordan elimination, each pivot is 1 and all entries
above and below each pivot are 0. Consequently, the row echelon form produced
by the Gauss–Jordan method contains a reduced number of nonzero entries, so
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it seems only natural to refer to this as a reduced row echelon form.

Reduced Row Echelon Form
A matrix E m×n is said to be in reduced row echelon form provided
that the following three conditions hold.
• E is in row echelon form.
• The first nonzero entry in each row (i.e., each pivot) is 1.
• All entries above each pivot are 0.
A typical structure for a matrix in reduced row echelon form is illustrated
below, where entries marked * can be either zero or nonzero numbers:

1
∗ 0 0 ∗∗ 0 ∗
 
 0 0 0 ∗∗ 0 ∗ 
1
 0 0 0 ∗∗ 0 ∗ 


1
 .
 0 0 0 0 0 0 ∗ 
1

0 0 0 0 0 0 0 0
 
0 0 0 0 0 0 0 0
As previously stated, if matrix A is transformed to a row echelon form
by row operations, then the “form” is uniquely determined by A, but the in-
dividual entries in the form are not unique. However, if A is transformed by
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row operations to a reduced row echelon form E A , then it can be shown that
both the “form” as well as the individual entries in E A are uniquely determined
by A. In other words, the reduced row echelon form E A produced from A is
independent of whatever elimination scheme is used. Producing an unreduced
form is computationally more efficient, but the uniqueness of E A makes it more
useful for theoretical purposes.
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In some of the older books this is called the Hermite normal form in honor of the French
mathematician Charles Hermite (1822–1901), who, around 1851, investigated reducing matrices
by row operations.
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A formal uniqueness proof must wait until Example 3.9.2, but you can make this intuitively
clear right now with some experiments. Try to produce two different reduced row echelon forms
from the same matrix.