Hessian Matrix

最新推荐文章于 2025-02-01 09:53:35 发布

seamanj

最新推荐文章于 2025-02-01 09:53:35 发布

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本文介绍了二阶偏导数检验方法，这是一种多元微积分中用于确定函数临界点性质的技术。通过分析临界点处的海森矩阵特征值，可以判断该点是局部最小值点、局部最大值点还是鞍点。

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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.