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