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Math Help - Euclidean Norm

  1. #1
    Junior Member
    Sep 2009
    Johannesburg, South Africa

    Euclidean Norm

    We have the Euclidean norm (or l_2-norm) for a vector on the vector space R^m ; \parallel{v}\parallel_2 = \sqrt{\sum_{j=1}^m\mid{v_j}\mid^2}.

    Given a vector norm ||\cdot|| on R^m the induced matrix norm for m x m matrices A is defined by
    ||A||\ =\ max_{v\neq{0}}\frac{||Av||}{||v||}
    That is , ||A|| is the smallest number \alpha such that  ||Av||\leq\alpha||v||\ \forall\ v \in R^m

    So given the Euclidian-norm for a vector the induced matrix norm ||A||_2=\sqrt{max.\ eigenvalue A^TA}
    Can anyone proof/explain why?
    In my book it is given but I donīt see it directly, i think it has something to do with the fact that A^TA is symmetric.
    Last edited by bram kierkels; October 4th 2009 at 08:02 AM.
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