Exercise 6.10.12

Answers

1.

A

and

B

are unitarily equivalent. We may write

B = Q^{∗} AQ

for some unitary matrix

Q

. Since

Q

is unitary, we have

∥ Q (x) ∥ = ∥ x ∥

. So we have

\frac{∥ Bx ∥}{∥ x ∥} = \frac{∥ Q^{∗} AQx ∥}{∥ x ∥} = \frac{∥ AQx ∥}{∥ Qx ∥} .

Since any unitary matrix is invertible, we get the equality $∥ A ∥ = ∥ B ∥$ .

2.

Write

β = {v_{1}, v_{2}, \dots, v_{n}}

and

x = \sum_{i = 1}^{n} a_{i} v_{i} .

We observe that

∥ x ∥^{2} = ⟨ x, x ⟩ = \sum_{i = 1}^{n} a_{i}^{2} = ∥ ϕ_{β} (x) ∥^{2},

where $ϕ_{β} (x)$ means the coordinates of $x$ with respect to $β$ . So we have

∥ x ∥ = ∥ ϕ_{β} (x) ∥

This means that

∥ T ∥ = \max_{x \neq 0} \frac{∥ T (x) ∥}{∥ x ∥} = \max_{x \neq 0} \frac{∥ ϕ_{β} (T (x)) ∥}{∥ ϕ_{β} (x) ∥}

= = \max_{ϕ_{β} (x) \neq 0} \frac{∥ {[T]}_{β} ϕ_{β} (x) ∥}{∥ ϕ_{β} (x) ∥} = ∥ {[T]}_{β} ∥ .

3.

We have

∥ T ∥ \geq k

for all integer

k

since we have

\frac{∥ T (v_{k}) ∥}{∥ v_{k} ∥} = \frac{∥ k v_{k} ∥}{∥ v_{k} ∥} = k .

jephianlin

2011-06-27 00:00