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Homework Help: Find an upper bound

  1. Jan 23, 2010 #1
    1. The problem statement, all variables and given/known data

    assume that x and y are vectors, and A is a matrix.

    can anyone kindly help me to find an upper bound C w.r.t [tex]\| A \| [/tex] s.t.

    [tex]\| x-Ay \| \leq C \cdot \| x-y\|[/tex]
     
  2. jcsd
  3. Jan 23, 2010 #2

    CompuChip

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    Some quick-and-dirty trial gives me
    C = sup( ||(A - I) v|| )
    where I is the identity matrix and the supremum is taken over all vectors v.

    I wonder if you can do any better, without more information on A.
     
  4. Jan 23, 2010 #3
    Thank you for ur kind help. if all the entries of A is between 0 and 1, can we get a nicer upper bound ?
     
  5. Jan 23, 2010 #4
    Could u show me any hints about ur estimate for C.

    I only figure out that [tex]\|x-Ay\|=\|(x-Ax)+(Ax-Ay)\|\leq\|I-A\| \|x\|+\|A\| \|x-y\|[/tex]

    I don't know how to continue... could anyone kindly give me more hints ?
     
    Last edited: Jan 24, 2010
  6. Jan 24, 2010 #5

    CompuChip

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    I did
    || x - A y|| = || (x - y) + (y - Ay) ||

    But when all entries of A are between 0 and 1, then you can define ||A|| by
    || A || = max(i, j)( |Aij| )
    and use that to get a better estimate.
     
  7. Jan 24, 2010 #6
    if || x - A y|| = || (x - y) + (y - Ay) ||,
    then || x - A y|| <= || (x - y) || + || (y - Ay) ||,

    but how could u find that C = sup |x-Ax| for all x ?

    notice that my esitmate is C*||x-y||
     
  8. Jan 25, 2010 #7
    can someone give me a hand?
     
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