Page 281 - Matrix Analysis & Applied Linear Algebra

5.1 Vector Norms 277

5.1.4. (a) Using the euclidean norm, describe the solid ball in n centered
at the origin with unit radius. (b) Describe a solid ball centered at
the point c =( ξ 1 ξ 2 ··· ξ n ) with radius ρ.

n T
5.1.5. If x, y ∈ such that x − y = x + y , what is x y?
2 2
5.1.6. Explain why x − y = y − x is true for all norms.

n
5.1.7. Forevery vector norm on C , prove that v depends continuously on
the components of v in the sense that for each > 0, there corresponds

a δ> 0 such that x − y < whenever |x i − y i | <δ for each i.

5.1.8. (a) For x ∈C n×1 , explain why x ≥ x ≥ x .
1 2 ∞
n×1
(b) For x ∈C , show that x ≤ α x , where α is the (i, j)-
i j
entry in the following matrix. (See Exercise 5.12.3 for a similar
statement regarding matrix norms.)

1 2 ∞
 √ 
1 ∗ n n
√
2  1 ∗ n  .
∞ 1 1 ∗

n
5.1.9. For x, y ∈C , x = 0, explain why equality holds in the CBS inequality
∗
∗
if and only if y = αx, where α = x y/x x. Hint: Use (5.1.4).
n
5.1.10. For nonzero vectors x, y ∈C with the euclidean norm, prove that
equality holds in the triangle inequality if and only if y = αx, where α
is real and positive. Hint: Make use of Exercise 5.1.9.

5.1.11. Use H¨older’s inequality (5.1.8) to prove that if the components of
n×1 T T
x ∈ sum to zero (i.e., x e =0 for e =(1, 1,..., 1) ), then

T y max − y min n×1
|x y|≤ x 1 for all y ∈ .
2
Note: For “zero sum” vectors x, this is at least as sharp and usually
it’s sharper than (5.1.8) because (y max − y min )/2 ≤ max i |y i | = y ∞ .