Given a vector , the Hessian of at is:
Here we view as a by matrix so that is the transpose of .
Remark. The Hessian of at is a quadratic form, since for any .
This is not difficult to show. Since is positive definite (http://planetmath.org/PositiveDefinite), the Rayleigh-Ritz theorem shows that there is a such that for all , . Thus by Taylor’s theorem (http://planetmath.org/TaylorPolynomialsInBanachSpaces) ( form)
For small the first on the the second, so that both sides are positive for small .
|Date of creation||2013-03-22 12:59:41|
|Last modified on||2013-03-22 12:59:41|
|Last modified by||cvalente (11260)|