Hessian matrix

Let xn and let f:n be a real-valued function having 2nd-order partial derivativesMathworldPlanetmath in an open set U containing x. The Hessian matrix of f is the matrix of second partial derivatives evaluated at x:

𝐇(x):=[2fx122fx1x22fx1xn2fx2x12fx222fx2xn2fxnx12fxnx22fxn2]. (1)

If f is in C2(U), 𝐇(x) is symmetricPlanetmathPlanetmath (http://planetmath.org/SymmetricMatrix) because of the equality of mixed partials. Note that 𝐇(x)=𝐉(f), the JacobianDlmfMathworldPlanetmathPlanetmath of the gradientMathworldPlanetmath of f.

Given a vector 𝒗n, the Hessian of f at 𝒗 is:

𝐇(x)(𝒗):=12𝒗𝐇(x)𝒗T. (2)

Here we view 𝒗 as a 1 by n matrix so that 𝒗T is the transposeMathworldPlanetmath of 𝒗.

Remark. The Hessian of f at 𝒗 is a quadratic formMathworldPlanetmath, since 𝐇(x)(r𝒗)=r2𝐇(x)(𝒗) for any r.

If f is further assumed to be in C2(U), and x is a critical pointDlmfMathworld of f such that 𝐇(x) is positive definitePlanetmathPlanetmath (http://planetmath.org/PositiveDefinite), then x is a strict local minimum of f.

This is not difficult to show. Since 𝐇(x) is positive definite (http://planetmath.org/PositiveDefinite), the Rayleigh-Ritz theorem shows that there is a c>0 such that for all hn, hT𝐇(x)h2ch2. Thus by Taylor’s theorem (http://planetmath.org/TaylorPolynomialsInBanachSpaces) ( form)


For small h the first on the the second, so that both sides are positive for small h.

Title Hessian matrix
Canonical name HessianMatrix
Date of creation 2013-03-22 12:59:41
Last modified on 2013-03-22 12:59:41
Owner cvalente (11260)
Last modified by cvalente (11260)
Numerical id 31
Author cvalente (11260)
Entry type Definition
Classification msc 26B12
Related topic Gradient
Related topic PartialDerivative
Related topic SymmetricMatrix
Related topic ComplexHessianMatrix
Related topic HessianForm
Related topic DirectionalDerivative
Defines Hessian