Hessian

Description

The Hessian is an object describing the “curvature” of a function. i.e. it describes how the differential of a function changes. In multivariate calculus we use it two determine extrema of functions.

Definition (Multivariate Calculus)

Let $U\subseteq {\mathbb{R}}^n$ and $f:U\to \mathbb{R}$. The Hessian matrix is defined as $(Hf)(a)=\left(\begin{array}{ccc}{\partial }_{x_1}{\partial }_{x_1}f(a)& ...& {\partial }_{x_1}{\partial }_{x_n}f(a)\\ ⋮& \ddots & ⋮\\ {\partial }_{x_n}{\partial }_{x_1}f(a)& ...& {\partial }_{x_n}{\partial }_{x_n}f(a)\end{array}\right)$ The theorem of Schwarz tells us that the Hessian is always symmetric.

Definition (Riemannian manifolds)

$Hess(f)(X,Y)=XYf-({\nabla }_{X}Y)f=({\nabla }_{X}df)(Y)$

The two concepts are actually the same. Both are bilinear forms, they take in two vectors and spit out a number. For the first definition, the vectors are multiplied left and right.

Properties

For multivariate calculus

Calculating extrema

Let $U\subseteq {\mathbb{R}}^n$ and $f:U\to \mathbb{R}$. We want to check if $a\in U$ is a local minimum or maximum. This can determined by the Hessian matrix $(Hf)(a)$

1. If $(Hf)(a)$ is negative definite, then we have an isolated maximum
2. If $(Hf)(a)$ is positive definite, then we have an isolated local minimum
3. If $(Hf)(a)$ is indefinite then we have a saddle point or a non-isolated extremum.

For Riemannian manifolds

Wert entlang Geodätischen

Sei $c:[a,b]\to B$ eine Lokale Geodätische (Mannigfaltigkeit). Dann gilt $Hess(f)(\dot{c},\dot{c})=(f\circ c{)}^{\prime \prime }$

Für ein $v\in T_{p}M$, realisiert durch $c$ gilt: $(Hessf)(v,v)=(f\circ c{)}^{\prime \prime }(0)=\Vert v{\Vert }^2$ wie man durch Einsetzen des Exponential (Geodätische) sehen kann.