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Does least squares solution to Ax=b depend on choice of norm

  1. Mar 6, 2013 #1
    To find the closest point to [itex]b[/itex] in the space spanned by the columns of [itex]A[/itex] we have:
    [tex]\mathbb{\hat{x}}=(A^TA)^{-1}A^T\mathbb{b}[/tex]
    My question is, shouldn't this solution ##\hat{x}## depend on the choice of distance function over the vector space? Choosing two different distance functions might give two different ##\hat{x}##s. But this equation does not make any reference to the choice of distance function.

    Can anyone explain this to me? This is not directly a homework question but I am just trying to get a better understanding of the concepts here.

    Thanks.
     
  2. jcsd
  3. Mar 6, 2013 #2

    mfb

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    Staff: Mentor

    If nothing else is given, use the standard scalar product and its induced norm.
     
  4. Mar 6, 2013 #3

    Ray Vickson

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    This function is *assuming* standard Euclidean distance. Of course for other measures of distance you will get different results. Some other norms do not lead to explicit solutions, but are only doable numerically.
     
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