Offensive Power Rankings for 2008

[Bongle]

Originally Posted by sumadin

I understand that p is what we're looking to find, or the average number of points a team contributes per match. s and M we can calculate from the results coming from each regional. Since Mp = s, we can conclude that p = (M-1)s (where M-1 denotes M inverse). It seems to me from thinking about it that M is symmetric, and as such M = M-1. Is that actually the case?

M is indeed symmetric. M[i][j] indicates how many matches team i played with team j. Therefore, it makes sense that M[i][j] = M[j][i].

I don't think the inverse of a symmetric matrix is necessarily symmetric. The transpose of a symmetric matrix M would be equal to its non-transpose, but I don't think that carries for symmetry. To prove that M^-1 is not necessarily equal to M for a symmetric matrix, just think of the symmetric matrix 2I, where I is the identity matrix.

If (M)^-1 = (M) for symmetric matrices like you are proposing, and we know MM^-1 = I for any matrix and its inverse, then (2I)(2I) must equal I, but this is not the case.

I didn't have to write the matrix solver myself, I used a library I found online. If there are any tricks it used, I'm not aware of them.