12 Apr
2016
12 Apr
'16
5:40 a.m.
Cool! That's a pretty subtle argument that doesn't look easy to discover. —Dan
On Apr 12, 2016, at 4:35 AM, Veit Elser <ve10@cornell.edu> wrote:
On Apr 12, 2016, at 12:50 AM, James Propp <jamespropp@gmail.com> wrote:
(6) On a more technical note, is there an easy way to see that the order of a Hadamard matrix must be 1, 2, or a multiple of 4?
Suppose the order is N>2. By multiplying rows by -1 you can always have the first three elements in each row be 1,1,1 or 1,1,-1 or 1,-1,1 or -1,1,1. Let the number of rows of these types be x,y,z,w. In addition to the constraint x+y+w+z=N, orthogonality of the first three columns gives three more constraints:
x+y-w-z=0 x-y-w+z=0 x-y+w-z=0
These imply x=y=z=w, and therefore N=x+y+z+w is a multiple of 4.