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Math Help - Cauchy-Schwarz Inequality

  1. #1
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    Cauchy-Schwarz Inequality

    If  a_{1}, \ldots, a_{n} and  b_{1} \ldots, b_{n} are arbitrary real numbers, then  \left(\sum_{k=1}^{n} a_{k}b_{k} \right)^{2} \leq \left(\sum_{k=1}^{n} a_{k}^{2} \right) \left(\sum_{k=1}^{n} b_{k}^{2} \right) . If some  a_{i} \neq 0 then equality holds if and only if there is a real  x such that  a_{k}x + b_{k} = 0 for each  k = 1,2, \ldots , n .

    So to basically prove this you do the following:  \sum_{k=1}^{n} (a_{k}x + b_{k})^{2} \geq 0  for all  x . It is equaled to  0 if  x_i = 0 .

    And this is equaled to  \sum_{k=1}^{n} a_{k}x^{2} + 2a_{k}x b_{k} + b_{k}^{2} \geq 0

    Then we invoke the discriminant of the quadratic equation  Ax^{2} + 2Bx + C \geq 0 by making the required substitutions?
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by heathrowjohnny View Post
    If  a_{1}, \ldots, a_{n} and  b_{1} \ldots, b_{n} are arbitrary real numbers, then  \left(\sum_{k=1}^{n} a_{k}b_{k} \right)^{2} \leq \left(\sum_{k=1}^{n} a_{k}^{2} \right) \left(\sum_{k=1}^{n} b_{k}^{2} \right) . If some  a_{i} \neq 0 then equality holds if and only if there is a real  x such that  a_{k}x + b_{k} = 0 for each  k = 1,2, \ldots , n .

    So to basically prove this you do the following:  \sum_{k=1}^{n} (a_{k}x + b_{k})^{2} \geq 0  for all  x . It is equaled to  0 if  x_i = 0 .

    And this is equaled to  \sum_{k=1}^{n} a_{k}x^{2} + 2a_{k}x b_{k} + b_{k}^{2} \geq 0

    Then we invoke the discriminant of the quadratic equation  Ax^{2} + 2Bx + C \geq 0 by making the required substitutions?

    have you ever tried searching from google or yahoo? in any field, the cauch-schwarz inequality have the same proof..

    correction:  \sum_{k=1}^{n} {a_{k}}^2x^{2} + 2a_{k}x b_{k} + b_{k}^{2} \geq 0

    anyways to continue your proof, yes, you must consider the discriminant, and that the discriminant must be \leq 0 (why?) (since  Ax^{2} + 2Bx + C \geq 0 , that is, if it is > 0, the equation has no real root and so the discriminant is < 0. ...)
    can you continue now??
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