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Math Help - Cauchy Schwarz question

  1. #1
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    Cauchy Schwarz question

    I have no idea to start. I'm struggling so much in this class. Can someone help me out?

    (a) Given that x² + y² + z² = 1, use the cauchy-schwarz inequality to find the largest possible value of the expression 2x + 3y + 6z

    (b) Let A be a diagonalizable matrix, all of whose eigenvalues are either 0 or 1. Show that A² = A

    thanks in advance!
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  2. #2
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    Suggestion

    If A = (2,3,6) and B = (x,y,z), then

    \vec A \cdot \vec B = 2x + 3y + 6z<br />
    is the expression you want to maximize.

    Cauchy Schwarz says
    (\vec A \cdot \vec B)^2 <= (\vec A \cdot \vec A) (\vec B \cdot \vec B)<br />

    B^2 = x^2 + y^2 + z^2 = 1<br />
    A^2 = 2^2 + 3^2 + 6^2 = 49<br />

    so
    (\vec A \cdot \vec B)^2 <= 49<br />

    or
    (\vec A \cdot \vec B) <= 7<br />

    For the 2nd, try
    A = S^{-1} D S<br />

    A^2 = S^{-1} D S S^{-1} D S = S^{-1} D^2 S = S^{-1} D S = A<br />

    since D has only diagonal elements 0 or 1.
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  3. #3
    Senior Member Shanks's Avatar
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    (b)Since A is a diagonalizable matrix, there exist invertable matrix B and diagonal matrix D( whose diagonal elements are 0 or 1), such that
    BAB^{-1}=D.
    since D^2=D, that is,
    BA^2B^{-1}=BAB^{-1}
    combined with B is invertable, gives the result.
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