Hadamard matrix question...

[Kyriakos]

http://mathworld.wolfram.com/HadamardMatrix.html

I was reading on this, and noticed that by definition those matrices must always have fewer white cells than black cells (namely for number of rows n, white cells will be n(n-1) and black cells will be n(n+1).

But if n=2 (ie the matrix has 8 rows, as in the image above) couldn't you just leave the very first top row with all white cells? Apparently this doesn't count, and i wanted to ask why.

I don't doubt the answer is simple, but if you can help it would save time for me. Moreover the actual conjecture (that Hadamard matrices/anallagmatic pavements only exist for n being 1, 2 or a multiple of 4) can be discussed

 

[red_elk]

But if n=2 (ie the matrix has 8 rows, as in the image above) couldn't you just leave the very first top row with all white cells? Apparently this doesn't count, and i wanted to ask why.

As far as I understand, this is just a rule of constructing these matrices invented by Sylvester.
Convention is that matrices have more positive elements than negative, and that positive elements correspond to black cells.
Under these rules you can't have top row filled with white cells, because the leftmost column contains only black cells.
Edit: Also it seems that all rows (except the bottom one) and columns (except left) must contain half of white and half of black cells.

But you can of course flip colors without violating the rule that neighboring rows and columns must be orthogonal to each other.
And rotate matrix, if you need

 

[Kyriakos]

^Oh... I didn't gather from the math article that both rows (top and every tier to the bottom) have to feature half same/half different color squares and columns (leftmost and everyone to the rightmost) must do the same as well!

Yes, that would entirely explain why all white top row doesn't work/isn't allowed

 

[red_elk]

On the other hand, if you need it for art purpose or something, you can construct matrix with top white row which would comply with orthogonality rule.

 

[Kyriakos]

No, i just need money, so i may lazily try to think about this issue for a while, until i no longer do