1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

2 questions related to norm

  1. Jul 12, 2007 #1
    Q1:
    how do we find a vector x so that ||A|| = ||Ax||/||x||(using the infinite norm)
    totally no clue on this question..

    Q2: Suppose that A is an n×n invertible matrix, and B is an approximation of A's inverse A^-1 such that AB = I + E for some matrix E. Show that the relative error in approximating A^-1 by B is bounded by ||E||(for some arbitrary matrix norm ||· ||).
    relative error=(a^-1-B)/A^-1
    AB=I+E--------1
    AA^-1=I-------2
    1/2: B/A^-1=1+E/I=1+E => (A^-1-B)/A^=-E
    now I'm stucking...how do I connect -E to norm of E?
    am I on the right track?any suggestion? thanks
     
  2. jcsd
  3. Jul 13, 2007 #2
    1) ||Ax|| is equal to the component of y=Ax with the maximum magnitude. Now each component of y satisfies

    [tex] y_i = A_i^Tx \leq \sum_j |a_{ij}|[/tex]

    for

    [tex]||x||_{\infty} \leq 1 [/tex]

    So the infinity norm of a matrix is

    [tex] \max_{i} \sum_j |a_{ij}| [/tex]

    Now it is straightforward to identify the desired x.


    2) Your syntax does not make any sense at all: What do you mean by B/A^-1? I suggest you follow matrix convention and use the definition for relative error.
     
    Last edited: Jul 13, 2007
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: 2 questions related to norm
  1. Confusing 2-norm proof (Replies: 6)

Loading...