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Gram–Schmidt process

  1. Aug 26, 2007 #1

    Can Gram–Schmidt process be used to orthonormalize a finite set of linearly independent vectors in a space with any nondegenerate sesquilinear form / symmetric bilinear form not necessarily positive definite?

    For example in R^2 define
    [tex]\langle a, b\rangle = - a_1\times a_1 + a_2\times a_2[/tex]

    From [tex]\{v_1, v_2\}[/tex] to [tex]\{e_1, e_2\}[/tex], assume v's are not null.
    [tex]e_1 = \frac{v_1}{|v_1|}[/tex]
    where [tex]|v_1| = \sqrt{|\langle v_1, v_1\rangle|}[/tex]
    [tex]t_2 = v_2 - \frac{\langle v_1, v_2\rangle}{\langle v_1, v_1\rangle}v_1[/tex]
    [tex]e_2 = \frac{t_2}{|t_2|}[/tex]

    It looks like it can be generalized to R^n without any problem.
    Last edited: Aug 26, 2007
  2. jcsd
  3. Aug 27, 2007 #2


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    In general, as long as you have a valid inner product, it works.
  4. Aug 27, 2007 #3
    Yes, I'm asking if we drop the assumption of positive definiteness of inner product, will it work?
  5. Aug 27, 2007 #4


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    well to get unit length vectors you divide by the length.
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