E^n中p维与q维线性子流形之间的距离

Distance between p-dimensional linear submanifold and q-dimensional linear submanifold in E^n

设n维欧氏空间En中p维与q维线性子流形分别为σp:︿1∧︿2∧…∧︿k∧(x-x0)=0,σq:﹀1∧﹀2∧…∧﹀q∧(y-y0)=0,向量组{︿1,…,︿p,﹀1,…,﹀q}的一个极大线性无关组为{︽1,︽2,…,︽k},证明了σp与σq间的距离平方为d2(σp,σq)=︾02-(︽1︾0,…,︽k︾0)A-1(︽1︾0,…,︽k︾0)T,其中 ︾0=x0-y0,A=︽i︽jki,j=1.

Let σp: α1∧α2∧…∧αp∧(x-x0)=0 and σq:β1∧β2∧…∧βq∧(y-y0)=0 are pdimensional linear submanifold and qdimensional linear submanifold in En respectively, then the square of distance between σp and σq isd2(σp,σq)=δ02-(γ1δ0,…,γkδ0)A-1(γ1δ0,…,γkδ0)T.Here,｛γ1,γ2,…,γk｝ is maximal linearly independent subset of ｛α1,…,αp,β1,…,βq｝,δ0=x0-y0, A=(γiγj)ki,j=1.