R^n中二次函数在单位球面上最值的探讨

ON MAXIMUM AND MINIMUM OF A QUADRATIC FUNCTION ON THE UNIT SPHERE IN R^n

1 问题的提出及其转化我们的问题是:设A为n阶实对称矩陈,r（A）=K（1≤K≤n）,a∈Rn为非零常向量,求f（X）=XT AX-2XT a在单位球面S={X|XT X=1,X∈Rn}上的最值。为简便起见,先将f（X）化简:作线性变换X=PY（P为n阶正交矩阵）,则 f（X）=XT AX-2XTa=YT PT APY-2YT PT a=YT AY-2YT c=φ（Y）。其中A=,λi（1≤i≤k）为矩阵A的谱,c=PT a=（c1,c2,…,cn）T

In this paper we study the analytic properties of a function ψ(λ) with Lagrange's method of multi piers and abtain that the maximal and minimal real roots of conditional equation g(λ)=1 are the maximun and minimun Points of ψ (λ), We also discuss the maximun and minimun problem of a quadric form on unit sphere,