## 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.