Math 2.0

Real functions Definition

Hessian matrix

Builds on Higher partial derivatives

Definition

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, let f ⁣:U→ℝf \colon U \to \mathbb{R}, and let a∈Ua \in U be a point at which all second partial derivatives of ff exist. The Hessian matrix of ff at aa is the n×nn \times n matrix Hf(a)=(∂2f∂xi ∂xj(a))1≤i,j≤n,H_f(a) = \left( \frac{\partial^2 f}{\partial x_i \, \partial x_j}(a) \right)_{1 \le i, j \le n}, whose entry in row ii and column jj is ∂2f/∂xi ∂xj\partial^2 f / \partial x_i \, \partial x_j at aa. It is also written ∇2f(a)\nabla^2 f(a) or D2f(a)D^2 f(a).

Remarks

If the second partial derivatives of ff are continuous near aa, then Hf(a)H_f(a) is a symmetric matrix, by Schwarz's theorem. For f(x,y)=x2+3xy−y2f(x, y) = x^2 + 3 x y - y^2, the Hessian matrix at each point is (233−2)\begin{pmatrix} 2 & 3 \\ 3 & -2 \end{pmatrix}. The Hessian matrix is the Jacobian matrix of the gradient map x↦∇f(x)x \mapsto \nabla f(x), when this map is differentiable. At a critical point, the signs of the eigenvalues of Hf(a)H_f(a) decide the second derivative test.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Hessian_matrix, Definitions and properties
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