Hessian matrix#Bordered Hessian

The rimmed Hessian matrix ( engl. bordered Hessian ) is used for classification of stationary points in multi-dimensional extreme value problems with constraints. It is related to the "normal" Hessian matrix. In contrast to the Hessian matrix, which is examined for a positive or negative definiteness in the rimmed Hessian matrix, the sign of the determinant is essential.

Decisive is the sign sequence of the leading principal minors, with the proviso that only the k leading principal minors examined for valid (m number of constraints ). If one examines, for example, a function for variables with a constraint, one must consider, therefore only the sign from the 3rd leading principal minor (See example below ).

Form ( 2-dimensional case )

For a two-dimensional function with a constraint which edged Hessian matrix has the following form.

Be the Lagrangian, with an arbitrary two-dimensional function and the constraint is to be optimized under which.

A stationary point of is then subject to the constraint

• Local maximum when
• Local minimum when
• Undecidable when