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Math Help - positive semi definite

  1. #1
    MHF Contributor chiph588@'s Avatar
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    positive definite matrix *Urgent*

    Show if K is positive definite then so is K^2
    Last edited by chiph588@; October 9th 2008 at 07:04 PM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by chiph588@ View Post
    Show if K is positive definite then so is K^2
    The matrix K is positive definite if for all real non-zeros vectors x:

    x^TKx>0

    K is positive definite, so consider:

    x^TK^2x

    for any non-zero real vector x, then:

    x^TK^2x=x^TKKx=\frac{x^TKxx^TKx}{|x|^2}=\frac{(x^T  Kx)(x^TKx)}{|x|^2}>0

    etc.

    CB
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  3. #3
    MHF Contributor chiph588@'s Avatar
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    why does  x^T KKx = \frac{x^T Kxx^T Kx}{|x|^2} ?
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by chiph588@ View Post
    why does  x^T KKx = \frac{x^T Kxx^T Kx}{|x|^2} ?
    Because x^Tx is a scalar equal to |x|^2

    CB
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