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Math Help - The maximum value of this quadratic form.

  1. #1
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    The maximum value of this quadratic form.

    Sorry if this isn't in quite the right place, but the question is a Linear Algebra question, even though I can't really relate it to linear algebra myself, bar the obvious quadratic form.

    Let a1,a2,...,an be real numbers such that:

    a1+a2+a3+...+an=0 and
    a1^2+a2^2+a3^2+...+an^2=1 (In case notation is ambiguous, 'a one squared plus ... plus a en squared = 1')

    Question:

    What is the maximum value of a1*a2+a2*a3+...+an*a1?

    I am completely stumped. No clue at all, not even how to begin (apart from treating all the a's as independent variables and differentiating, but that gives me a horrible mess when it comes to put in the constraints). Any help would be appreciated.

    Also, sorry about lack of LATEX knowhow.
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  2. #2
    Super Member
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    Try starting with this:

    Since \sum_{i=1}^{n}a_n^2 = 1, we get that for each 1 \leq i \leq n, ~ a_i^2 \leq 1 (can you see why?).

    Now, you try to continue from here.
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  3. #3
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    Quote Originally Posted by Defunkt View Post
    Try starting with this:

    Since \sum_{i=1}^{n}a_n^2 = 1, we get that for each 1 \leq i \leq n, ~ a_i^2 \leq 1 (can you see why?).

    Now, you try to continue from here.
    Each of them is less than or equal to 1 else the sum of the squares would be greater than 1. This tells me that the sum I'm looking for has an absolute bound above of n. But, it can never attain this as it would require all of the variables to equal 1, so I don't see what I'm supposed to do with it...
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