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

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Page 1: 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

http://myhome.spu.edu/lauw

Page 2: MAT 4725 Numerical Analysis Section 7.1 (Part II) Norms of Vectors and Matrices

Test Maple

Page 3: MAT 4725 Numerical Analysis Section 7.1 (Part II) Norms of Vectors and Matrices

7.1 Norms of Vectors and Matrices

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

Page 4: MAT 4725 Numerical Analysis Section 7.1 (Part II) Norms of Vectors and Matrices

Spaces of nxn Matrices

Identify 2

real matricesn n n R

Page 5: MAT 4725 Numerical Analysis Section 7.1 (Part II) Norms of Vectors and Matrices

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

Page 6: MAT 4725 Numerical Analysis Section 7.1 (Part II) Norms of Vectors and Matrices

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

Page 7: MAT 4725 Numerical Analysis Section 7.1 (Part II) Norms of Vectors and Matrices

l2 Norm

22 21

maxx

A Ax

Page 8: MAT 4725 Numerical Analysis Section 7.1 (Part II) Norms of Vectors and Matrices

l2 Norm 22 21

maxx

A Ax

Page 9: MAT 4725 Numerical Analysis Section 7.1 (Part II) Norms of Vectors and Matrices

l Norm

1maxx

A Ax

Page 10: MAT 4725 Numerical Analysis Section 7.1 (Part II) Norms of Vectors and Matrices

Theorem 7.11

11

maxn

iji n

j

A a

Come Back

1. . max i

i nc f x x

Page 11: MAT 4725 Numerical Analysis Section 7.1 (Part II) Norms of Vectors and Matrices

Example 1

5 1 6

4 1 0

4 10 2

?

A

A

Page 12: MAT 4725 Numerical Analysis Section 7.1 (Part II) Norms of Vectors and Matrices

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

Page 13: MAT 4725 Numerical Analysis Section 7.1 (Part II) Norms of Vectors and Matrices

Equivalent Definition for the Natural Matrix Norm

0max

z

AzA

z

Page 14: MAT 4725 Numerical Analysis Section 7.1 (Part II) Norms of Vectors and Matrices

Equivalent Definition for the Natural Matrix Norm

1max

xA Ax

0max

z

AzA

z

Page 15: MAT 4725 Numerical Analysis Section 7.1 (Part II) Norms of Vectors and Matrices

Corollary 7.10

If is a vector norm and 0 then

z

Az A z

Page 16: MAT 4725 Numerical Analysis Section 7.1 (Part II) Norms of Vectors and Matrices

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.

Page 17: MAT 4725 Numerical Analysis Section 7.1 (Part II) Norms of Vectors and Matrices

Homework

Download HW Read Section 7.2