Matrix norms

  • Thread starter meemoe_uk
  • Start date
Duh, I can`t calculate matrix norms using the formula....

||A|| = max || Ax || where || x || = 1

This is how I try to calculate them, what am I doing wrong?

e.g. Find norm 2 of A

A = 1 1
0 1

First find A's eigen system....
Solve characteristic polynomial....
( 1 - k ) ( 1 - k )
k = 1 - eigen value of A
Get eigen vector....
A - k I = 0
0 1 = 0
0 0 = 0

eigen vector = 1
0

As || Ax || is at a maximum when x is A's eigen vector, we can now calculate ||A||.
Ax = 1 1 * 1 = 1
= 0 1 0 = 0
Therefore
|| A || = || 1 || = 1
|| 0 ||

Actual answer = 1.618

Bah. I can do it for norm 1 and infinity, but not any number inbetween. I'm not allowed to use that traspose matrix ,spectral radius formula. What's the secret??? Please help.

I can`t seem to display a matrix nicely on my post either. sos
 
Last edited:

Hurkyl

Staff Emeritus
Science Advisor
Gold Member
14,829
14
Hrm.

Are you sure you have the right matrix? Based on the correct answer, my guess is that it's supposed to be

Code:
A = /0 1\
    \1 1/
 
Hi Hurkyl,
The matrix is the one from Burden - Faires Numerical Analysis 4th Edition Ex Set 7.2 Q 1 b)
If I use the spectral radius formula I get the right answer.

Here' another eg. Q 1 d)

A =
2 1 1
2 3 2
1 1 2

Solve characteristic polynomial
- k^3 + 7k^2 - 11k + 5
( k - 1 ) ^2 ( k - 5 )
k = 1 , 5

Get eigenvectors
For k = 1
A - kI = 0 =
1 1 1
2 2 2
1 1 1

solution space vectors =
1
-1
0

1
0
-1

For k = 5
A - kI = 0 =
-3 1 1
2 -2 2
1 1 -3

solution space vector =
1
2
1

|| Ax || is at maximum when x is eigen vector corisponding to largest eigen value so k=5 and
x =
1
2
1
/ Sqr 6 , to nomalize || x || = 1

Calculate Ax

2 1 1 * 1
2 3 2 * 2
1 1 2 * 1 / Sqr 6

=
5
10
5 / Sqr 6

Get Norm...
= Sqr ((25 + 100 + 25) / 6)
= Sqr ( 150 / 6 )
= 5 My answer

Actual Answer = 5.2035

I get the eigen system correct, but it's the matrix norm calculation where I go wrong I think.
 

Want to reply to this thread?

"Matrix norms" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Top Threads

Top