Page 51 - Matrix Analysis & Applied Linear Algebra

44 Chapter 2 Rectangular Systems and Echelon Forms

Notice that the final result of applying Gaussian elimination in the above
example is not a purely triangular form but rather a jagged or “stair-step” type
of triangular form. Hereafter, a matrix that exhibits this stair-step structure will
be said to be in row echelon form.

Row Echelon Form
An m × n matrix E with rows E i∗ and columns E ∗j is said to be in
row echelon form provided the following two conditions hold.

• If E i∗ consists entirely of zeros, then all rows below E i∗ are also
entirely zero; i.e., all zero rows are at the bottom.

• If the first nonzero entry in E i∗ lies in the j th position, then all
entries below the i th position in columns E ∗1 , E ∗2 ,..., E ∗j are zero.

These two conditions say that the nonzero entries in an echelon form
must lie on or above a stair-step line that emanates from the upper-
left-hand corner and slopes down and to the right. The pivots are the
first nonzero entries in each row. A typical structure for a matrix in row
echelon form is illustrated below with the pivots circled.
 
* ∗ ∗ ∗ ∗ ∗ ∗ ∗
 0 0 * ∗ ∗ ∗ ∗ ∗ 
 0 0 0 * ∗ ∗ ∗ ∗ 


 0 0 0 0 0 0 * ∗ 


0 0 0 0 0 0 0 0
 
0 0 0 0 0 0 0 0
Because of the flexibility in choosing row operations to reduce a matrix A
to a row echelon form E, the entries in E are not uniquely determined by A.
Nevertheless, it can be proven that the “form” of E is unique in the sense that
the positions of the pivots in E (and A) are uniquely determined by the entries
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in A . Because the pivotal positions are unique, it follows that the number of
pivots, which is the same as the number of nonzero rows in E, is also uniquely
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determined by the entries in A . This number is called the rank of A, and it
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The fact that the pivotal positions are unique should be intuitively evident. If it isn’t, take the
matrix given in (2.1.1) and try to force some different pivotal positions by a different sequence
of row operations.
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The word “rank” was introduced in 1879 by the German mathematician Ferdinand Georg
Frobenius (p. 662), who thought of it as the size of the largest nonzero minor determinant
in A. But the concept had been used as early as 1851 by the English mathematician James
J. Sylvester (1814–1897).