Results 1 to 2 of 2

Thread: Vector Norms

  1. #1
    Newbie
    Joined
    Feb 2010
    Posts
    5

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Feb 2010
    From
    Lisbon
    Posts
    51
    You just need to check the axioms for a norm. Let $\displaystyle x,y\in \mathbb R^n$, and $\displaystyle \lambda \in \mathbb R$. Then


    Positive definiteness:

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


    Positive homogeneity:

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


    Triangle inequality:

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


    Tadaa!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. norms on R
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Oct 10th 2011, 02:42 PM
  2. Norms of a general vector space
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Aug 24th 2011, 05:07 PM
  3. Norms
    Posted in the Advanced Applied Math Forum
    Replies: 2
    Last Post: Jun 18th 2011, 03:07 PM
  4. Equivalence of vector norms
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Jul 3rd 2010, 09:44 PM
  5. vector norms and matrix norms
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: May 13th 2010, 02:42 PM

Search Tags


/mathhelpforum @mathhelpforum