Page 141 - Matrix Analysis & Applied Linear Algebra

3.9 ElementaryMatrices and Equivalence 135

Furthermore, by permuting the rows of E 1 and E 2 to force the pivotal 1’s to
occupy the diagonal positions, we see that
row row
E 1 ∼ T 1 and E 2 ∼ T 2 , (3.9.5)
where T 1 and T 2 are upper-triangular matrices in which the basic columns in
each T i occupy the same positions as the basic columns in E i . For example, if
   
120 120
E =  001  , then T =  000  .
000 001

2
Each T i has the property that T = T i because there is a permutation
i
matrix Q i (a product of elementary interchange matrices of Type I) such that

T
Q i T i Q = I r i J i or, equivalently, T i = Q T I r i J i Q i ,
i
i
0 0 0 0
2
and Q T = Q −1 (see Exercise 3.9.4) implies T = T i . It follows from (3.9.5)
i
i
i
row
that T 1 ∼ T 2 , so there is a nonsingular matrix R such that RT 1 = T 2 . Thus
and T 1 = R −1 T 2 = R −1 T 2 T 2 = T 1 T 2 .
T 2 = RT 1 = RT 1 T 1 = T 2 T 1
Because T 1 and T 2 are both upper triangular, T 1 T 2 and T 2 T 1 have the same
diagonal entries, and hence T 1 and T 2 have the same diagonal. Therefore, the
positions of the basic columns (i.e., the pivotal positions) in T 1 agree with those
in T 2 , and hence E 1 and E 2 have basic columns in exactly the same positions.
This means there is a permutation matrix Q such that

I r J 1 I r J 2
E 1 Q = and E 2 Q = .
0 0 0 0
Using (3.9.4) yields PE 1 Q = E 2 Q, or

P 11 P 12 I r J 1 I r J 2
= ,
P 21 P 22 0 0 0 0
which in turn implies that P 11 = I r and P 11 J 1 = J 2 . Consequently, J 1 = J 2 ,
and it follows that E 1 = E 2 .
In passing, notice that the uniqueness of E A implies the uniqueness of the
row
pivot positions in any other row echelon form derived from A. If A ∼ U 1
row
and A ∼ U 2 , where U 1 and U 2 are row echelon forms with different pivot
positions, then Gauss–Jordan reduction applied to U 1 and U 2 would lead to
two different reduced echelon forms, which is impossible.
In §2.2 we observed the fact that the column relationships in a matrix A
are exactly the same as the column relationships in E A . This observation is a
special case of the more general result presented below.