On Wednesday 25 May 2005 16:18, Mike Speciner wrote:
If by summetry you mean the fact that all but one of the rows have the same number of 0s as 1s, you can replace 10011011 with 00011011 and have them all have that property. (The cofactor of the changed bit is 0.) You'll note that the columns are not particularly symmetric in this sense.
No, what I meant was the presence of things like 11001100 and 01010101 and 01100110, which have considerably more symmetry to them than just equal numbers of 0s and 1s.
You haven't described your hillclimbing algorithm, but to maximize the determinant while changing a single row, you clearly set each element in the row to 0 if its cofactor is negative and 1 if the cofactor is positive. (If the cofactor is 0, the element doesn't matter.) If you assume that a cofactor is as likely to be positive as negative, then you'd expect roughly the same number of 0s as 1s.
The hillclimbing algorithm was the simplest imaginable: try all possible single-bit changes and choose a random maximal-determinant result; repeat until no further improvement is possible. Your observation about cofactors is astute, though I suspect (with no evidence) that global optima might be more accessible by a hillclimbing approach that takes fewer steps and leaves more open to chance. (But I'd have done the cofactor thing if I'd thought of it.) In another message you mention starting with random matrices and discarding ones with negative determinant; of course you can just transpose two rows if you start with a negative determinant :-). -- g