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Thread: 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@; Oct 9th 2008 at 06:04 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 $\displaystyle K$ is positive definite if for all real non-zeros vectors $\displaystyle x$:

    $\displaystyle x^TKx>0$

    $\displaystyle K$ is positive definite, so consider:

    $\displaystyle x^TK^2x$

    for any non-zero real vector $\displaystyle x$, then:

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

    etc.

    CB
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  3. #3
    MHF Contributor chiph588@'s Avatar
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    why does $\displaystyle 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 $\displaystyle x^T KKx = \frac{x^T Kxx^T Kx}{|x|^2} $?
    Because $\displaystyle x^Tx$ is a scalar equal to $\displaystyle |x|^2$

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