Description
1. Given two vectors u, v, ∈ R
n
, and real scalars α, β, let A = I + αuvT
, B = I + βuvT
.
(a) If u, v, α are given, find β such that B = A−1
.
(b) For which values of α is A singular, if any? For that particular value of α, give a
non-zero vector x in the right nullspace of A. Write x in terms of u, v, α.
(c) Prove of disprove: for any given pair of vector u, v, there always exists a value α
such that A singular. To prove, show such an α always exists, giving a formula in
terms of u, v. To disprove, give an example of a pair of non-zero vectors u, v for
which no such α exists. In the latter case, what general property do u, v satisfy to
prevent the existence of α? You can illustrate your answer with a 2 × 2 example.
(d) Give a value of α (in terms of u, v) such that A2 = A (i.e., A is a projector).
[Hint: Multiply out (I + αuvT
)(I + βuvT
) and find value for β to reduce the product to
the desired result.]
2. Let fp(v) = maxkukp=1 |u
T
v|, where kvkp denotes the p-norm.
(a) Prove or disprove: fp is a vector norm. (check each property, or show one is violated).
(b) Give a formula for fp for p = 1, 2. Hint, the answers can be written in terms of
k · k2, k · k∞. For p = 2, use the Cauchy-Schwartz inequality.
3. Define the inner product among square matrices by hA, Bi = trace(ATB), where A, B are
n × n matrices.
(a) What is the norm induced by this inner product: kAk
2 = hA, Ai? Answer this
question for the general case for any A.
• Now answer the remaining questions below using this specific matrix:
A =
6 −2 1
7 −7 3
−4 5 −2
.
(b) For this specific matrix A, what is the value of hA, Ai and the corresponding induced
norm kAk =
q
hA, Ai from part (a)?
(c) What is the p norm of A, for p = 1? Find a vector x s.t. kxkp = 1 and kAkp =
kAxkp.
(d) Repeat the above for p = 2. Use Matlab and write the result to 4 decimal places.
Show your Matlab commands.
(e) Use Matlab to help solve this problem: Find a vector x achieving the minimum in
minkxkp=1 kAxkp. Do this for p = 1, 2.
Note: in this exercise, you can use Octave instead of Matlab.
