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Math Help - need to know insight concept or underlaying concept of Cauchy–Schwarz inequality

  1. #1
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    need to know insight concept or underlaying concept of Cauchy–Schwarz inequality

    CAN ANYONE PLEASE TELL ME ABOUT
    Cauchy–Schwarz inequality
    1. WHAT IT MEANS ACTUALLY I KNOW FORMAL DEFINITION BUT I WANA INSIGHT CONCEPT WHY PRODUCT OF INDIVIDUAL VECTORS IS ALWAYS> AND EQUAL THAN INNER PRODUCT

    2. SPECIFY THE CASE WHEN PRODUCT OF MAGNITUDES OF IND.VECTORS IS = MAG OF INNER PRODUCT
    MAY THIS HELPS ME TO GET INSIGHT CONCEPT

    I AM NEW TO THESES FORUMS IF ANYONE CAN EMAIL ME IT WILL BE GREAT FOR ME...
    THANKX.
    E-MAIL ADRESS:
    junaidanwar194@gmail.com
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  2. #2
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    Re: need to know insight concept or underlaying concept of Cauchy–Schwarz inequality

    Suppose you have two UNIT vectors \hat{v}, \hat{w} in some inner product space (V, <,>).

    Then \vec{z} =  <\hat{v}, \hat{w}> \hat{w} is the component of \hat{v} in the \hat{w} direction.

    (Recall that \hat{v} = \vec{z} + (\hat{v} - \vec{z}), decomposes \hat{v} into parallel and perpendicular components w/resp to \hat{w}:

    <(\hat{v} - \vec{z}), \hat{w}> = <\hat{v}, \hat{w}> - <\vec{z}, \hat{w}> = <\hat{v}, \hat{w}> - <<\hat{v}, \hat{w}> \hat{w}, \hat{w}>

    = <\hat{v}, \hat{w}> - <\hat{v}, \hat{w}><\hat{w}, \hat{w}> = <\hat{v}, \hat{w}>( 1 - <\hat{w}, \hat{w}>) = 0.)

    Conceptually, should \hat{v}, some generic unit vector, have a greater component in the \hat{w} direction than \hat{w} itself does?

    No. Conceptually, each unit vector "should have the greatest magnitude among all unit vector in its own direction." Right?

    OK. So that means we expect.... \lVert \vec{z} \rVert \le \lVert \hat{w} \rVert = 1, with equality iff \hat{v} = \pm \hat{w}.

    Thus we expect.... \lVert <\hat{v}, \hat{w}> \hat{w} \rVert \le 1, with equality iff \hat{v} = \pm \hat{w}.

    Thus we expect.... \lvert <\hat{v}, \hat{w}> \rvert \lVert \hat{w} \rVert \le 1, with equality iff \hat{v} = \pm \hat{w}.

    Thus we expect.... \lvert <\hat{v}, \hat{w}> \rvert \le 1, with equality iff \hat{v} = \pm \hat{w}.

    Thus we expect inner products of *unit* vectors to take values in [-1,1], with it being \pm1 only when one vector is \pm1 the other.


    Now, suppose \vec{a} and \vec{b} are any two non-zero vectors (CS obviously holds if one of the vectors is \vec{0}). What do we expect?

    Consider the proceeding in light of their unit vectors \frac{\vec{a}}{\lVert \vec{a} \rVert} and \frac{\vec{b}}{\lVert \vec{b} \rVert}:

    We expect.... \left\lvert \left<\frac{\vec{a}}{\lVert \vec{a} \rVert}, \frac{\vec{b}}{\lVert \vec{b} \rVert}\right> \right\rvert \le 1, with equality iff \frac{\vec{a}}{\lVert \vec{a} \rVert} = \pm \frac{\vec{b}}{\lVert \vec{b} \rVert}, i.e. iff \vec{a} = \left( \pm \frac{\lVert \vec{a} \rVert }{\lVert \vec{b} \rVert } \right)\vec{b}.

    So we expect.... \lvert <\vec{a}, \vec{b}> \rvert \le \lVert \vec{a} \rVert \ \lVert \vec{b} \rVert, with equality iff \vec{a} = some multiple of \vec{b}.

    Thus we expect the Cauchy-Schwarz inequality to hold - because we expect inner products of unit vectors to take values in [-1,1].

    That's its conceptual meaning to me.
    Last edited by johnsomeone; October 1st 2012 at 10:25 AM.
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