Page 132 - Matrix Analysis & Applied Linear Algebra

126 Chapter 3 Matrix Algebra

Another sum that often requires inversion is I − A, but we have to be
careful because (I−A) −1 need not always exist. However, we are safe when the
entries in A are sufficiently small. In particular, if the entries in A are small
n
enough in magnitude to insure that lim n→∞ A = 0, then, analogous to scalar
algebra,

2
n
(I − A)(I + A + A + ··· + A n−1 )= I − A → I as n →∞,
so we have the following matrix version of a geometric series.

Neumann Series
n
If lim n→∞ A = 0, then I − A is nonsingular and
∞
k
−1
2
(I − A) = I + A + A + ··· = A . (3.8.5)
k=0
This is the Neumann series. It provides approximations of (I − A) −1
when A has entries of small magnitude. For example, a first-order ap-
proximation is (I − A) −1 ≈ I+A. More on the Neumann series appears
in Example 7.3.1, p. 527, and the complete statement is developed on
p. 618.

While there is no useful formula for (A + B) −1 in general, the Neumann
series allows us to say something when B has small entries relative to A, or
vice versa. For example, if A −1 exists, and if the entries in B are small enough
n
in magnitude to insure that lim n→∞ A −1 B = 0, then

−1 −1
(A + B) −1 = A I − −A −1 B = I − −A −1 B A −1

∞
k −1

−1
= −A B A ,
k=0
and a first-order approximation is
(A + B) −1 ≈ A −1 − A −1 BA −1 . (3.8.6)

Consequently, if A is perturbed by a small matrix B, possibly resulting from
errors due to inexact measurements or perhaps from roundoff error, then the
resulting change in A −1 is about A −1 BA −1 . In other words, the effect of a
small perturbation (or error) B is magnified by multiplication (on both sides)
with A −1 , so if A −1 has large entries, small perturbations (or errors) in A can
produce large perturbations (or errors) in the resulting inverse. You can reach