# Hessian

Summary

• If $$f$$ is a function of $$2$$ variables then the $$2×2$$ matrix

$H=\begin{pmatrix} f''_{xx}(x,y) & f''_{xy}(x,y) \\ f''_{yx}(x,y) & f''_{yy}(x,y) \end{pmatrix}$

• is called the Hessian of $$f$$ .
• If $$f$$ is a smooth function then $$f_{xy}^{''}\left( x,y \right)=f_{yx}^{''}\left( x,y \right)$$ and the Hessian is a symmetric matrix.
• The determinant of the Hessian , denoted by $$|H|$$ is

$|H|=f_{xx}^{''}f_{yy}^{''}-f_{yx}^{''}f_{xy}^{''}$

• which, in general, is a function of $$x$$ and $$y$$ .
• We say that the Hessian is:
• Positive semidefinite if and only if $$f_{xx}^{''}≥0$$ and $$|H|≥0$$
• Negative semidefinite if and only if $$f_{xx}^{''}≤0$$ and $$|H|≥0$$
• Positive definite if and only if $$f_{xx}^{''}>0$$ and $$|H|>0$$
• Negative definite if and only if $$f_{xx}^{''}<0$$ and $$|H|>0$$
• for all $$x,y$$ in the domain of $$f$$ .

Convex and concave functions

• We say that $$A⊆R^2$$ is a convex set if the following condition holds: Pick any two points in $$A$$ . If all points along a straight line between these points are in $$A$$ then we say that $$A$$ is a convex set.
• $$f:A→R^2$$ is a function of $$2$$ variables where $$A$$ is a convex subset of $$R^2$$ and $$f$$ is twice differentiable. $$H$$ is the Hessian of $$f$$ . Then
• $$H$$ is positive semidefinite on $$A$$ $$\iff$$ $$f$$ is convex on $$A$$
• $$H$$ is negative semidefinite on $$A$$ $$\iff$$ $$f$$ is concave on $$A$$
• $$H$$ is positive definite on $$A$$ $$⇒$$ $$f$$ is strictly convex on $$A$$ (opposite not necessarily true)
• $$H$$ is negative definite on $$A$$ $$⇒$$ $$f$$ is strictly concave on $$A$$ (opposite not necessarily true)
• Example: $$f(x,y)=x^2+y^2$$ on $$R^2$$ has a Hessian which is positive definite for all $$(x,y)$$ and $$f$$ is strictly convex on $$R^2$$ .