推 KJLP :感謝前輩! 02/11 19:13
※ 引述《KJLP (PWKJ)》之銘言：
: 1)A is a k×k matrix all of whose entries are ±1
: and whose rows are mutually orthogonal.
: Suppose that A has an m×n submatrix whose entries are all 1.
: show that mn≦k.
: 實在沒頭緒的一題 感覺應該是要利用A的每列orthogonal
: 以及子矩陣每列相同這點?
By condition, there are distinct rows v1,v2,..,vm
of the form vi=(1,1,...,1, ui)
--------- n
then <vi,vj> = n + <ui,uj> = kd_ij, where d_ij=0 if i!=j,
1 if i=j.
Taking sum over all 1<= i,j <=m,
nm^2 + ||u1+..+um||^2 = km
so nm^2 <= km, nm<=k.
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