从Lagrange恒等式到Hadamard不等式  

From Lagrange Identity to Hadamard Inequality

作　　者：周久儒[1] ZHOU Jiu-ru(School of Mathematical Sciences, Yangzhou University, Yangzhou Jiangsu 225002, China)

机构地区：[1]扬州大学数学科学学院

出　　处：《大学数学》2019年第3期111-114,共4页College Mathematics

基　　金：江苏省高校品牌专业建设工程资助项目(PPZY2015B109);江苏省高校自然科学项目(15KJB110024)

摘　　要：从简单等式(a2+b2)(c2+d2)=(ac+bd)2+(ad-bc)2,联想到Lagrange恒等式,并借用Grassmann代数得到该等式的多变量推广.考虑一般的Lagrange恒等式的几何意义,得到关于k个n维向量张成的平行多面体体积的恒等式.作为应用,分别得出CauchyGSchwartz不等式和Hadamard不等式的几何证明.This paper starts from a simple identity (a 2+b 2)(c 2+d 2)= (ac+bd) 2+ (ad-bc) 2 , to the Lagrange identity, and generalizes this identity to multiple variables by Grassmann algebra. Considering the geometric meanings of the generalized Lagrange identity, we get an identity involving the volume of a parallelohedron generated by k n -dimensional vectors. As applications, we obtain geometric proofs of Cauchy-Schwartz inequality and Hadamard inequality.

关 键 词：Lagrange恒等式 GRASSMANN代数 CauchyGSchwartz不等式 HADAMARD不等式 

分 类 号：O183[理学—数学]