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Math Help - Vector Norms

  1. #1
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    Vector Norms

    Let S be a real and nonsingular matrix, and let ||.|| be any norm on R^n. Define ||.||' by ||x||' = ||Sx||. Show that ||.||' is also a norm on R^n.

    I know ||S|| = max ||Sx|| from ||x||=1 but I'm not sure if that applies.
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  2. #2
    Junior Member
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    You just need to check the axioms for a norm. Let x,y\in \mathbb R^n, and \lambda \in \mathbb R. Then


    Positive definiteness:

    \|x\|'=0\Leftrightarrow \| S\cdot x\| =0
    \Leftrightarrow S\cdot x=0 (since \|\cdot\| is a norm)
    \Leftrightarrow x=0 (since S is nonsingular, and thus invertible)


    Positive homogeneity:

    \|\lambda\cdot x\|'=\|S\cdot (\lambda\cdot x)\| = \|\lambda \cdot (S\cdot x)\|
    =|\lambda|\cdot \|S\cdot x\| (for \|\cdot\| is a norm)
    =|\lambda|\cdot \|x\|'


    Triangle inequality:

    \|x+y\|'=\|S\cdot (x+y)\|=\| S\cdot x+S\cdot y\|
    \leq \|S\cdot x\|+\|S\cdot y\| (again, \|\cdot\| is a norm)
    =\|x\|'+\|y\|'


    Tadaa!
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