剖析二次型理论在多元函数极值和条件极值中的应用

Analysis of the Application of Quadratic Form Theory in Extremum and Conditional Extremum of Multivariate Function

摘要: 二次型理论是高等代数中的一个重要分支，其在多元函数极值和条件极值中具有重要的地位。由于二次型理论涉及到代数学、分析学等方面的知识，因此，对学生而言学习难度较大。实际上，运用矩阵的正定性来判断多元函数的极值和条件极值，是一种极为便捷的方法，但其适用范围存在一定的限制。因为驻点处对应的矩阵的正定、负定仅是该点成为极值点的充分条件。鉴于此，本文首先探讨了多元函数的极值与矩阵特征值之间的相互联系，其次深入研究二次型理论在多元函数极值、最值问题中的应用。同时还讨论了二次型理论在多元函数条件极值中的应用及其存在的不足。当判别法失效时，通常需要在驻点处判断Lagrange函数的二阶微分的符号。通过这几方面的介绍，旨在加深学生对该理论的理解和应用。

Abstract: Quadratic theory is an important branch of advanced algebra, and it plays an important role in the extreme value and conditional value of multivariate function. Because quadratic theory involves algebra and analytic knowledge, it is difficult for students to learn. In fact, it is an extremely con-venient method to determine the extremum and conditional extremum of multivariate function by the positive properties of matrix, but its application scope is limited. Because the positive definite and negative definite of the matrix corresponding to the stagnation point are only sufficient con-ditions for the point to become the extreme point. In view of this, this paper first discusses the correlation between the extreme value of multivariate function and the eigenvalue of matrix, and then studies the application of quadratic theory in the extreme value and optimal value of multi-variate function. At the same time, the application of quadratic theory in conditional extreme value of multivariate function and its shortcomings are discussed. When the discriminant fails, it is usually necessary to judge the sign of the second derivative of Lagrange function at the stagnation point. Through the introduction of these aspects, the aim is to deepen students’ understanding and application of the theory.