[TBag]

So what about this,

since we're doing dot products and the matrix a is always a diagonal matrix of 1's or -1's, either going from top left to bottom right or bottom left to top right.
IE
[(-)1 0 0 : 0 (-)1 0 : 0 0 (-)1 ],
[0 0 (-)1 : 0 (-1) 0 : (-1) 0 0 ], etc

We can assume one of two things is happening.

1)
Ax = (A11 * x1) + (A22 * x2) ... (Ann * xn)
and same goes for y, replacing the x

or if the ones line up from bottom left to top right
2)
Ax = (A1n * xn) + (A2(n-1) * x(n-1) ) .... (An1 * x1)
and same goes for y,

so regardless of which path we take, in Ax (dot) Ay, we end up with (either A entry)^2(x1)(y1)+ ... + A^2(xn)(yn) which is the same as regular x (dot) y because any A entry squared will be 1.