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Math Help - Prove matrx is positive definite

  1. #1
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    Prove matrx is positive definite

    Show that matrix A =
    |a b|
    |0 c|

    is positive definite if a > 0, c > 0 and |b| < 2(ac)^.5

    not really sure how to prove this, i multiplied an vector z = [x,y]^T so that

    z^T * A * z = ax^2 + byx + cy^2 > 0 but i am not sure is this is even the right direction.
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  2. #2
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    Quote Originally Posted by beachbump View Post
    Show that matrix A =
    |a b|
    |0 c|

    is positive definite if a > 0, c > 0 and |b| < 2(ac)^.5

    not really sure how to prove this, i multiplied an vector z = [x,y]^T so that

    z^T * A * z = ax^2 + byx + cy^2 > 0 but i am not sure is this is even the right direction.

    The expression ax^2+byx+cy^2 is a quadratic in x with positive quadratic coefficient and thus it will ALWAYS be positive iff its discriminant is negative:

    \Delta = b^2y^2-4acy^2<0\Longleftrightarrow b^2-4ac<0\Longleftrightarrow |b|<2\sqrt{ac}

    Tonio
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