Linear algebra/ optimization proof

[SNOOTCHIEBOOCHEE]

Homework Statement

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

The Attempt at a Solution

Im having trouble with this problem because i don't know enough about linear albegra.

What types of matrices have negative eigenvalues? is there some sort of identity that I am 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 d^T \nabla ^2f(x)d will be less that zero but i have no clue how to make that intial statement.

 

[Dick]

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.