What is the maximum |determinant| of an NxN circulant matrix whose defining first row consists entirely of +1's and -1's? (Related question: each entry lies between -1 and +1 inclusive?) -------- The answer is <=N^(N/2), but this upper bound is achieved only when N=1 and N=4 (by -1,1,1,1). When N=5 the best matrix is (-1,1,1,1,1). Thus the sequence begins 1,0,4,16,48 and the next entry perhaps is 128 from (-1,1,1,1,1,1). The NxN circulant matrix (-1,1,1,...,1) has det = N * (-2)^(N+1) = -1,0,+4,-16,+48,-128... but this formula definitely stops yielding maximum |det| at N=7 because formula would yield 320 whereas the matrix (-1,1,-1,-1,1,1,1) has det=512. [Possible calculational tricks: The first entry is +1 wlog saving you a factor of 2. The second entry is >= the last entry, saving further factor. The determinant of an NxN circulant can be calculated by taking the FFT of the first row, scaling, then product of FFT entries, see http://en.wikipedia.org/wiki/Circulant_matrix which runs in O(NlogN) operations. Using however the slow discrete FT instead of fast FT we can improve to time O(N) per matrix by examining matrices in gray-code order.] -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)