Results 1 to 3 of 3

Math Help - Optimization proof...

  1. #1
    Junior Member ginafara's Avatar
    Joined
    May 2007
    Posts
    26

    Optimization proof...

    Can anyone help me with with this proof???

    Show that a sufficient condition for...

    Lim as k-> infinity, A^K = 0

    is

    ||A|| < 1 (NOTE: this is the 2-norm)

    any help would be greatly appreciated...
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by ginafara View Post
    Can anyone help me with with this proof???

    Show that a sufficient condition for...

    Lim as k-> infinity, A^K = 0

    is

    ||A|| < 1 (NOTE: this is the 2-norm)

    any help would be greatly appreciated...
    || A^K ||_2 = max_{x !=0} [||A^K x||_2 / ||x||_2]

    ................ = max_{x!=0}|| [A ( A^{K-1} x||_2 / ||X||_2]

    ................ < k max_{x!=0}|| [ A^{K-1} x||_2 / ||X||_2]

    for some k such that: ||A||_2 < k < 1, hence:

    || A^K ||_2 < k^K

    But as K -> infty, k^K ->0, so

    Lim as k-> infinity, ||A^K||_2 = 0

    hence A^K -> 0 (as in this case convergence in norm implies convergence elementwise
    though I'm not sure if you will have to prove this).

    RonL
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member ginafara's Avatar
    Joined
    May 2007
    Posts
    26
    beautiful... thanks for the help... I was on the right track, but wasn't confident in my work.


    beautiful proof... thanks again
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Continuous Optimization: Proof of Convexity
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: January 15th 2012, 09:49 PM
  2. Replies: 5
    Last Post: October 19th 2010, 10:50 AM
  3. Optimization
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 21st 2008, 01:53 PM
  4. proof that the proof that .999_ = 1 is not a proof (version)
    Posted in the Advanced Applied Math Forum
    Replies: 4
    Last Post: April 14th 2008, 04:07 PM
  5. Yet More Optimization
    Posted in the Calculus Forum
    Replies: 3
    Last Post: November 15th 2006, 04:23 PM

Search Tags


/mathhelpforum @mathhelpforum