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Math Help - Positive Definiteness

  1. #1
    Junior Member
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    Positive Definiteness

    Hi,

    I'm trying to show that the following matrix:

    A = \begin{bmatrix} 1&2&3\\2&5&1\\3&1&36 \end{bmatrix} is positive definite. Using the typical definitions I get to this inequality:

    x^2 + 2x*(2y+3z) + 5y^2 + 2*y*z + 36z^2 > 0. I just don't know to show that this is true.

    Also, I'm trying to show that this matrix:

    B = \begin{bmatrix} 1&2&3\\2&5&1\\3&1&34 \end{bmatrix} is NOT positive definite. I can't find a vector x that shows x^T*B*x > 0.

    Thanks a lot for your help.
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  2. #2
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    Never mind
    Last edited by dwsmith; December 9th 2010 at 04:48 PM. Reason: Part of definition
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  3. #3
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    Quote Originally Posted by AKTilted View Post
    Hi,

    I'm trying to show that the following matrix:

    A = \begin{bmatrix} 1&2&3\\2&5&1\\3&1&36 \end{bmatrix} is positive definite. Using the typical definitions I get to this inequality:

    x^2 + 2x*(2y+3z) + 5y^2 + 2*y*z + 36z^2 > 0. I just don't know to show that this is true.

    Also, I'm trying to show that this matrix:

    B = \begin{bmatrix} 1&2&3\\2&5&1\\3&1&34 \end{bmatrix} is NOT positive definite. I can't find a vector x that shows x^T*B*x > 0.

    Thanks a lot for your help.

    Dou you know Sylvester's Criterion? A is pos. def. because all its principal minors are positive, whereas B doesn't fulfill this

    condition (in fact, \det B =0 ).

    So in order to find an element x\in\mathbb{R}^3\,\,s.t.\,\,x^tBx\ngtr 0 , just choose a non-trivial vector in the

    kernel of B...

    Tonio
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