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Operator norm upper bound

  1. Apr 16, 2013 #1
    Greetings everyone!

    I have a set of tasks I need to solve using using operator norms, inner product... and have some problems with the task in the attachment. I would really appreciate your help and advice, since my deadline is tomorrow.

    This is what I have been thinking about so far:
    I have to calculate a non trivial upper bound, so maybe it could be done by:
    [tex] b=max( ||A||_1,||A||_2,||A||_\infty ) [/tex]

    Since [tex] A [/tex] is a difference operator I estimated the following:
    [tex] ||A||_1= 4 [/tex]
    [tex] ||A||_\infty= 2 [/tex]
    But how can I estimate [tex]||A||_2=?[/tex]
    If I know that abs row sum is 2 (besides 0 there appears only one 1 and one -1 in the rows) and abs column sum is 4 (it is two times the size of row lenght dim(A)=2mn x mn). Can I estimate [tex]||A||_2[/tex] by:
    [tex]||A||_2=\sqrt{rows^2+columns^2 }=\sqrt{(2 \cdot 2mn)^2+(4 \cdot mn)^2}[/tex][tex]=4 \sqrt{(mn)^2+(mn)^2}=4 \sqrt{2} \sqrt{m^2n^2}[/tex] since [tex]mn[/tex] are positive [tex]||A||_2=4 \sqrt{2} mn[/tex]
    So I would say [tex]b=max(L_1,L_2,L_\infty)=L_2=4 \sqrt{2} mn[/tex]

    Is my conclusion, approximation of a non trivial upper bound b right?

    Thank you in advance for your help!
     

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    Last edited: Apr 16, 2013
  2. jcsd
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