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Matrix norms

  1. Aug 4, 2003 #1
    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: Aug 4, 2003
  2. jcsd
  3. Aug 4, 2003 #2

    Hurkyl

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    Staff Emeritus
    Science Advisor
    Gold Member

    Hrm.

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

    Code (Text):

    A = /0 1\
        \1 1/
     
     
  4. Aug 5, 2003 #3
    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.
     
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