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Math Help - What is a Norm? Norm of a vector? P-Norm?

  1. #1
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    Exclamation What is a Norm? Norm of a vector? P-Norm?

    so norm of a number gives out the length
    norm of a vector gives out the magnitude
    but what about p norm?
    what is a p norm and what does it solve for?
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  2. #2
    Super Member craig's Avatar
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    This link may be of help to you

    Matrix Norm -- from Wolfram MathWorld
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by bnr34rb26dett View Post
    so norm of a number gives out the length
    norm of a vector gives out the magnitude
    but what about p norm?
    what is a p norm and what does it solve for?
    \| \bold{x} \|_p = \left( \sum_i |x_i|^p \right)^{1/p}

    CB
    Last edited by CaptainBlack; December 10th 2009 at 06:39 AM.
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  4. #4
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    In general, a "norm" on a vector space is a function that assigns to each vector a number, ||v||, satifying:
    1) ||u+ v|| \le ||u||+ ||v||.
    2) ||v||= 0 if and only if v is the 0 vector.
    3) ||av||= |a|||v|| for any scalar (number) a, and vector v.

    The formula Captain Black gives defines the "p" norm, over R^n, for any positive number p (p is usually taken to be an integer but that is not necessary)- though I believe it should include an absolute value:
    ||v||= \left(\sum |x_i|^p\right)^{1/p}.

    Of course, the p-norm for p= 2 is the usual "Euclidean norm" ||v||= \sqrt{\sum x_i^2}.

    If p= 1 we get the "one-norm" ||v||= \sum |x_i|.

    Sometimes, although it doesn't fit the formula above, we define the "0-norm" to be ||v||= max |x_i|- that is, take the absolute value of all components, then select the largest to be the norm.
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