mat 4725 numerical analysis section 7.1 (part ii) norms of vectors and matrices

MAT 4725Numerical Analysis

Section 7.1 (Part II)

Norms of Vectors and Matrices

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Test Maple

7.1 Norms of Vectors and Matrices

Norms on real vector space (Part I) Norms on Matrices (Part II)

Spaces of nxn Matrices

Identify 2

real matricesn n n R

Definition 7.82 2

2

2

2

2

A on is a function : s.t.

(i) 0

(ii) 0 iff

(iii) ,

(iv) + ,

(v) ,

n n

n

n

n

n

A A

m

A A

A A A

A B A B

atrix n

A B

AB A B A B

orm

0

R R R

R

R R

R

R

Theorem 7.9 (Natural Matrix Norm)

2

1

If is a vector norm then

max

is a matrix norm ( , )

x

n n

A Ax

A x

R R

HW

l2 Norm

22 21

maxx

A Ax

l2 Norm 22 21

maxx

A Ax

l Norm

1maxx

A Ax

Theorem 7.11

11

maxn

iji n

j

A a

Come Back

1. . max i

i nc f x x

Example 1

5 1 6

4 1 0

4 10 2

?

A

A

Example 1

5 1 6

4 1 0

4 10 2

?

A

A

11

maxn

iji n

j

A a

1

1

2

3

1

1

n

jj

n

jj

n

jj

a

a

a

A

Equivalent Definition for the Natural Matrix Norm

0max

z

AzA

z

Equivalent Definition for the Natural Matrix Norm

1max

xA Ax

0max

z

AzA

z

Corollary 7.10

If is a vector norm and 0 then

z

Az A z

Classwork

Prove Theorem 7.11. Step by step instructions are given.

Work in a group of 2 If you do not like “n”, you can work with

“3” first. Reverse approach – For part 1, you may

do part (c) first.

Homework

Download HW Read Section 7.2