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Linear algebra/ optimization proof

  1. Jun 3, 2009 #1
    1. The problem statement, all variables and given/known data

    A vector d is a direction of negative curvature for the function f at the point x if dT [tex]\nabla ^2[/tex]f(x)d <0. Prove that such a direction exists if at least one of the eigenvalues of [tex]\nabla ^2[/tex] f(x) is negative


    3. The attempt at a solution

    Im having trouble with this problem because i dont know enough about linear albegra.

    What types of matrices have negative eigenvalues? is there some sort of identity that im missing? can somebody point me in the right direction?

    Basically i think this proof is going to go like somehow having a negative eigenvalue implies that dT [tex]\nabla ^2[/tex]f(x)d will be less that zero but i have no clue how to make that intial statement.
     
    Last edited by a moderator: Jun 3, 2009
  2. jcsd
  3. Jun 3, 2009 #2

    Dick

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    You hardly need to know anything. A matrix A has an eigenvalue L if there is a vector v such that Av=Lv. v^Tv>=0 (it's just v.v, the dot product). Just substitute your operator for A.
     
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