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2 questions related to norm

  1. Jul 12, 2007 #1
    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
    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]


    [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
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