从Lagrange恒等式到Hadamard不等式

From Lagrange Identity to Hadamard Inequality

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.