Chapter 3: Inner Products and Norms

Question 1

Prove that || • || is a continuous function

Answer 1

See chapter 3 notes

Question 2

Prove that:

1/sqr(n) || v ||_{<span>2</span>} <= || v ||₁ <= || v ||₂

over V = R^n for any norms || ||₁ and || ||₂

Answer 2

See notes

Question 3

Let V be a normed vector space and T be a linear operator on V. Prove that T is continuous at all points in V.

Answer 3

See notes

Question 4

|| A ||_\infty is a natural matrix norm

Answer 4

True

Question 5

A unit sphere in V is enough to compute the matrix norm

|| A ||

Answer 5

True

Question 6

Prove that a narural matrix norm is finite.

Answer 6

Hint: unit sphere is finite and T is continuous.

Question 7

Prove that:

|| Av || <= || A || || v ||

Answer 7

See notes

Question 8

|| A B || <= || A || || B ||

Answer 8

True

Question 9

|| A ||_\infty = maximal absolute column sum of A

Answer 9

False

Question 10

|| A ||₁ = maximal absolute columns sum of A

Answer 10

True

Question 11

Let A ~ B, i.e.

S^-1AS = B

Then ||A|| = ||B||.

Answer 11

False

Question 12

Let ||A||_F = 5, ||B||_F= 6. Then ||AB||_F can have which of the following values?

• 30
• 31
• 32
• 33

Answer 12

30

Question 13

K is Positive Definite implies K is symmetric?

Answer 13

False. This is only true for Real matrices.

Question 14

K PD implies K = conj (K^T)

Answer 14

True for all F

Question 15

K = ((1 0), (0 -1)) defines an inner product

Answer 15

False. Only PD matrices define an inner product. K is indefinite

Question 16

Can two distinct PD matrices K_1 and K_2 define the same inner product?

Answer 16

Yes. If K_1 and K_2 are the matrices of the same opetrator w.r.t. different bases

Question 17

K = ((1 0), (0 -1)) admits a null direction?

Answer 17

True. x = (a, a) is a null direction for all real a

Question 18

Every non-singular (invertible) matrix is P.D.

Answer 18

False

Question 19

Every P.D. Matrix is non-singular

Answer 19

True

Question 20

Suppose K is a real, non-singular, square matrix. Then (K^T)^-1 = (K^-1)^T ?

Answer 20

True

Question 21

K be a diagonal matrix with non-zero entries. Then K must be P.D.

Answer 21

False. e.g. K = ((1 0), (0 -1)) is not P.D.

Question 22

K is a diagonal matrix with positive entries, then K is P.D.

Answer 22

True

Question 23

K is a diagonal matrix which is P.D. Then K can have a zero entry.

Answer 23

False. That would make K non-invertible.