Mathematics-Online lexicon: Hesse Matrix

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	Mathematics-Online lexicon:
	Hesse Matrix

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overview

Let be a scalar function with continuous second partial derivatives. The symmetric matrix

$\displaystyle \operatorname{H}f(a) = \left( \begin{array}{ccc} \partial_1\part... ...ial_n\partial_1 f(a) & \cdots & \partial_n\partial_n f(a) \end{array}\right)$

is called the Hesse matrix of at The Hesse matrix consists of the second partial derivatives of

The expansion of in a Taylor series till partial derivatives of second order has with the Hesse matrix the form

$\displaystyle f(x-a) = f(a) + \left(\operatorname{grad}f(a)\right)^{\operatorna... ...a) + \frac{1}{2} (x-a)^t \operatorname{H}f (a) (x-a) + O(\vert(x-a)\vert^3) .$

()

Annotation:

Beweis: Darstellung der Taylorentwicklung mit der Hesse-Matrix (german)

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automatically generated 8/ 4/2008