Page 142 - Matrix Analysis & Applied Linear Algebra

136 Chapter 3 Matrix Algebra

Column and Row Relationships

row
• If A ∼ B, then linear relationships existing among columns of A
also hold among corresponding columns of B. That is,

n n

B ∗k = α j B ∗j if and only if A ∗k = α j A ∗j . (3.9.6)
j=1 j=1
• In particular, the column relationships in A and E A must be iden-
tical, so the nonbasic columns in A must be linear combinations of
the basic columns in A as described in (2.2.3).
col
• If A ∼ B, then linear relationships existing among rows of A must
also hold among corresponding rows of B.
• Summary. Row equivalence preserves column relationships, and col-
umn equivalence preserves row relationships.

row
Proof. If A ∼ B, then PA = B for some nonsingular P. Recall from (3.5.5)
that the j th column in B is given by
B ∗j =(PA) ∗j = PA ∗j .

Therefore, if A ∗k = α j A ∗j , then multiplication by P on the left produces
j
B ∗k = α j B ∗j . Conversely, if B ∗k = α j B ∗j , then multiplication on the
j j
left by P −1 produces A ∗k = α j A ∗j . The statement concerning column
j
equivalence follows by considering transposes.
The reduced row echelon form E A is as far as we can go in reducing A by
using only row operations. However, if we are allowed to use row operations in
conjunction with column operations, then, as described below, the end result of
a complete reduction is much simpler.

Rank Normal Form

If A is an m × n matrix such that rank (A)= r, then

I r 0
A ∼ N r = . (3.9.7)
0 0
N r is called the rank normal form for A, and it is the end product
of a complete reduction of A by using both row and column operations.