实对称矩阵在求多元函数极值中的应用

The application of symmetry matrix when solve the extremism value of several-variables function

设点P(a1,a2,…,an)是n元函数f(X)=f(x1,x2,…,xn)的一个稳定点,当P有增量ΔP=(h1,h2,…hn)时,相应地函数有增量Δf=f(P+ΔP)-f(P).根据Δf的不同情况,可以判断f(P)是不是极值,是极大值还是极小值.由泰勒(Taylor)公式及高阶无穷小的概念知道,Δf的主要组成部分是一个关于h1,h2,…,hn的实二次型,其系数为f(X)在点P处对各自变量的二阶偏导数和二阶混合偏导数,其矩阵是一个实对称矩阵,用A表示,如果A为正定矩阵,则二次型为正定二次型,Δf>0,从而f(P)为极小值;如果A为负定矩阵,则二次型为负定二次型,Δf<0,从而f(P)为极大值;如果A既不正定,又不负定,则f(P)不是极值.

If is a stable point of an n - variables function, and if point P gets an addition, the relative function also obtains an addition . In the different cases of , we can judge if is an extremism value： either maximized value or minimized value. Based on the definition of the Taylor formula and the infinitesimal, we know that the main part of is a real - quadratic function about while its coefficient is the second - order partial derivatives at the point P of , and its matrix is a real - symmetric matrix called A. If A is a positively definite matrix that the quadratic function is a positively definite quadratic function and , so is a minimized value. If A is a negatively definite matrix that the quadratic function is a negatively definite quadratic function and , so is a maximized. When A is neither positively defined nor negatively defined, is not an extremism value.