Page 220 - Matrix Analysis & Applied Linear Algebra

4.5 More about Rank 215

solution. However if 3-digit arithmetic is used to form the associated system of
normal equations, the result is

10 20 x 1 30
= .
20 40 x 2 60.1
T
The 3-digit representation of A A is singular, and the associated system of
normal equations is inconsistent. For these reasons, the normal equations are
often avoided in numerical computations. Nevertheless, the normal equations
are an important theoretical idea that leads to practical tools of fundamental
importance such as the method of least squares developed in §4.6 and §5.13.

Because the concept of rank is at the heart of our subject, it’s important to
understand rank from a variety of different viewpoints. The statement below is
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one more way to think about rank.

Rank and the Largest Nonsingular Submatrix

The rank of a matrix A m×n is precisely the order of a maximal square
nonsingular submatrix of A. In other words, to say rank (A)= r
means that there is at least one r × r nonsingular submatrix in A,
and there are no nonsingular submatrices of larger order.

Proof. First demonstrate that there exists an r × r nonsingular submatrix in
A, and then show there can be no nonsingular submatrix of larger order. Begin
with the fact that there must be a maximal linearly independent set of r rows
in A as well as a maximal independent set of r columns, and prove that the
submatrix M r×r lying on the intersection of these r rows and r columns is
nonsingular. The r independent rows can be permuted to the top, and the
remaining rows can be annihilated using row operations, so

row U r×n
A ∼ .
0
Now permute the r independent columns containing M to the left-hand side,
and use column operations to annihilate the remaining columns to conclude that

row U r×n col M r×r N col M r×r 0
A ∼ ∼ ∼ .
0 0 0 0 0
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This is the last characterization of rank presented in this text, but historically this was the
essence of the first definition (p. 44) of rank given by Georg Frobenius (p. 662) in 1879.