Hessian Matrix

引用于
https://en.wikipedia.org/wiki/Second_partial_derivative_test#Functions_of_many_variables

In mathematics, the second partial derivative test is a method in multivariable calculus used to determine if a critical point of a function is a local minimum, maximum or saddle point.

For a function f of tow or more variables, to determine what kind of point a critical point is, one must look at the eigenvalues of the Hessian matrix at the critical point. The following test can be applied at any critical point (a, b, …) for which the Hessian matrix is invertible:

If the Hessian is positive definite (equivalently, has all eigenvalues positive) at (a, b, …), then f attains a local minimum at (a, b, …).
If the Hessian is negative definite (equivalently, has all eigenvalues negative) at (a, b, …), then f attains a local maximum at (a, b, …).
If the Hessian has both positive and negative eigenvalues then (a, b, …) is a saddle point for f (and in fact this is true even if (a, b, …) is degenerate).

In those cases not listed above, the test is inconclusive.