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Hessian matrix positive/negative definite Watch

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    Hi,

    Having trouble wrapping my head around working out whether a hessian matrix is positive, positive semi, negative or negative semi definite.

    E.g. a 2x2 matrix:

    a11 a12
    a21 a22

    am I right in thinking it

    is positive definite if a11 > 0 and the determinant > 0 (minimum)
    is negative definite if a11 < 0 and determinant is > 0 (maximum)
    indefinite if the determinant is < 0 (saddle point)

    Have no idea how to check for semi definite though, if anyone could help or correct me it would be much appreciated
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    (Original post by Scorcher)
    Hi,

    Having trouble wrapping my head around working out whether a hessian matrix is positive, positive semi, negative or negative semi definite.

    E.g. a 2x2 matrix:

    a11 a12
    a21 a22

    am I right in thinking it

    is positive definite if a11 > 0 and the determinant > 0 (minimum)
    is negative definite if a11 < 0 and determinant is > 0 (maximum)
    indefinite if the determinant is < 0 (saddle point)

    Have no idea how to check for semi definite though, if anyone could help or correct me it would be much appreciated
    No. The matrix {{5, 17}, {0, 11}} is indefinite: b = (1, -\frac{17}{22} + \frac{\sqrt{69}}{22}) gives b^T M b = 0, but det M is 55.

    Possibly the quickest way to work out these things: a real matrix is positive/negative definite iff the symmetric matrix \frac{1}{2} (M+M^T) is pos/neg def. A symmetric matrix is pos/neg def iff all its eigenvalues are positive/negative.
 
 
 
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