Page 136 - Matrix Analysis & Applied Linear Algebra

130 Chapter 3 Matrix Algebra

3.8.3. Suppose the coefficient matrix of a nonsingular system Ax = b is up-
T
dated to produce another nonsingular system (A + cd )z = b, where
n×1
b, c, d ∈ , and let y be the solution of Ay = c. Show that
T
T
z = x − yd x/(1 + d y).
3.8.4. (a) Use the Sherman–Morrison formula to prove that if A is non-
singular, then A + αe i e T is nonsingular for a sufficiently small
j
α.
(b) Use part (a) to prove that I + E is nonsingular when all - ij ’s
are sufficiently small in magnitude. This is an alternative to using
the Neumann series argument.

3.8.5. For given matrices A and B, where A is nonsingular, explain why
A + -B is also nonsingular when the real number - is constrained to
a sufficiently small interval about the origin. In other words, prove that
small perturbations of nonsingular matrices are also nonsingular.

3.8.6. Derive the Sherman–Morrison–Woodbury formula. Hint: Recall Exer-
I C A C I 0
cise 3.7.11, and consider the product T T .
0 I D −I D I
3.8.7. Using the norm (3.8.7), rank the following matrices according to their
degree of ill-conditioning:
   
100 0 −100 1 8 −1
A =  0 100 −100  , B =  −9 −71 11  ,
−100 −100 300 1 17 18
 
1 22 −42
C =  0 1 −45  .
−45 −948 1
3.8.8. Suppose that the entries in A(t), x(t), and b(t) are differentiable
functions of a real variable t such that A(t)x(t)= b(t).
(a) Assuming that A(t) −1 exists, explain why
dA(t) −1 −1 −1

= −A(t) A (t)A(t) .
dt
(b) Derive the equation

b (t) − A(t)
x (t)= A(t) −1 −1 A (t)x(t).
This shows that A −1 magnifies both the change in A and the
change in b, and thus it confirms the observation derived from
(3.8.8) saying that the sensitivity of a nonsingular system to
small perturbations is directly related to the magnitude of the
entries in A −1 .