Page 223 - Matrix Analysis & Applied Linear Algebra

218 Chapter 4 Vector Spaces

If the rank is not to jump, then the perturbation E must be such that S = 0,
−1
which is equivalent to saying E 22 = E 21 (I + E 11 ) E 12 . Clearly, this requires
the existence of a very specific (and quite special) relationship among the entries
of E, and a random perturbation will almost never produce such a relation-
ship. Although rounding errors cannot be considered to be truly random, they
are random enough so as to make the possibility that S = 0 very unlikely.
Consequently, when A is singular, the small perturbation E due to roundoff
makes the possibility that rank (A + E) > rank (A) very likely. The moral is
to avoid floating-point solutions of singular systems. Singular problems can often
be distilled down to a nonsingular core or to nonsingular pieces, and these are
the components you should be dealing with.

Since no more significant characterizations of rank will be given, it is ap-
propriate to conclude this section with a summary of all of the different ways we
have developed to say “rank.”

Summary of Rank
m×n
For A ∈ , each of the following statements is true.
• rank (A) = The number of nonzero rows in any row echelon form
that is row equivalent to A.
• rank (A) = The number of pivots obtained in reducing A toarow
echelon form with row operations.

• rank (A) = The number of basic columns in A (as well as the num-
ber of basic columns in any matrix that is row equivalent
to A ).

• rank (A) = The number of independent columns in A —i.e., the size
of a maximal independent set of columns from A.
• rank (A) = The number of independent rows in A —i.e., the size of
a maximal independent set of rows from A.

• rank (A) = dim R (A).
T
• rank (A) = dim R A .
• rank (A)= n − dim N (A).
T
• rank (A)= m − dim N A .
• rank (A) = The size of the largest nonsingular submatrix in A.
m×n T ∗
For A ∈C , replace (') with (') .