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Mathematics-Online lexicon:

Hesse Matrix

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

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

is called the Hesse matrix of $ f$ at $ a .$ The Hesse matrix consists of the second partial derivatives of $ f .$

The expansion of $ f$ 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)
. $



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  automatically generated 8/ 4/2008