Editing Hessian matrix (section)

== Definitions and properties ==

Suppose <math>f : \R^n \to \R</math> is a function taking as input a vector <math>\mathbf{x} \in \R^n</math> and outputting a scalar <math> f(\mathbf{x}) \in \R.</math> If all second-order [[partial derivative]]s of <math>f</math> exist, then the Hessian matrix <math>\mathbf{H}</math> of <math>f</math> is a square <math>n \times n</math> matrix, usually defined and arranged as 
<math display=block>\mathbf H_f= \begin{bmatrix}
  \dfrac{\partial^2 f}{\partial x_1^2} & \dfrac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_1\,\partial x_n} \\[2.2ex]
  \dfrac{\partial^2 f}{\partial x_2\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_2^2} & \cdots & \dfrac{\partial^2 f}{\partial x_2\,\partial x_n} \\[2.2ex]
  \vdots & \vdots & \ddots & \vdots \\[2.2ex]
  \dfrac{\partial^2 f}{\partial x_n\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_n^2}
\end{bmatrix}.</math>
That is, the entry of the {{mvar|i}}th row and the {{mvar|j}}th column is
<math display=block>(\mathbf H_f)_{i,j} = \frac{\partial^2 f}{\partial x_i \, \partial x_j}.</math>

If furthermore the second partial derivatives are all continuous, the Hessian matrix is a [[symmetric matrix]] by the [[symmetry of second derivatives]].

The [[determinant]] of the Hessian matrix is called the {{em|Hessian determinant}}.<ref>{{cite book|last1=Binmore|first1=Ken|author-link1=Kenneth Binmore|last2=Davies|first2=Joan|year=2007|title=Calculus Concepts and Methods|oclc=717598615|isbn=978-0-521-77541-0|publisher=Cambridge University Press|page=190}}</ref>

The Hessian matrix of a function <math>f</math> is the [[Jacobian matrix]] of the [[gradient]] of the function <math>f</math>; that is: <math>\mathbf{H}(f(\mathbf{x})) = \mathbf{J}(\nabla f(\mathbf{x})).</math>