Hessian Matrix - Mixed Derivatives and Symmetry of The Hessian

Mixed Derivatives and Symmetry of The Hessian

The mixed derivatives of f are the entries off the main diagonal in the Hessian. Assuming that they are continuous, the order of differentiation does not matter (Clairaut's theorem). For example,

\frac {\partial}{\partial x} \left( \frac { \partial f }{ \partial y} \right) = \frac {\partial}{\partial y} \left( \frac { \partial f }{ \partial x} \right).

This can also be written as:

In a formal statement: if the second derivatives of f are all continuous in a neighborhood D, then the Hessian of f is a symmetric matrix throughout D; see symmetry of second derivatives.

Read more about this topic:  Hessian Matrix

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