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Linear Algebra

Exercise 7.2.20

Answers

1.

The norm

∥ A ∥ \geq 0

since

| A_{ij} | \geq 0

for all

i

and

j

. The norm

∥ A ∥ = 0

if and only if

| A_{ij} | = 0

for all

i

and

j

. So

∥ A ∥ = 0

if and only if

A = O

.

2.

Compute

∥ cA ∥ = \max {| c A_{ij} |} = | c | \max {| A_{ij} |} = | c | ∥ A ∥ .

3.

It comes from the fact

| A_{ij} + B_{ij} | \leq | A_{ij} | + | B_{ij} |

for all $i$ and $j$ .

4.

Compute

∥ AB ∥ = \max {| {(AB)}_{ij} |} = \max {| \sum_{k = 1}^{n} A_{ik} B_{kj} |}

\leq \max {| \sum_{k = 1}^{n} ∥ A ∥ ∥ B ∥ |} = n ∥ A ∥ ∥ B ∥ .

jephianlin

2011-06-27 00:00

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