Hmm, these look interesting when expressed in binary: 1 10 100 10000 110000 10100000 1001000000 101000000000 11000000000000 10010000000000000 1010000000000000000 1000000000000000000000 111000111101000000000000 100000110101010000000000000 The challenge is to work out why this should occur. Sincerely, Adam P. Goucher ----- Original Message ----- From: "Warren Smith" <warren.wds@gmail.com> To: <math-fun@mailman.xmission.com> Sent: Saturday, August 18, 2012 8:37 PM Subject: [math-fun] NxN Toeplitz matrices with entries +-1 and maximum|determinant|
(This is a related, but different, problem from the previous one for circulants.)
A Toeplitz matrix is constant along diagonals. An NxN Toeplitz matrix with entries +-1 and maximum |determinant| is as follows, where I regard the Toeplitz as specified by its entries going up the first column, then rightward along the top row (i.e. "clockwise").
N det example matrix 1 1.00 + 2 2.00 +-- 3 4.00 --+++ 4 16.00 +---+-- 5 48.00 -++++-+++ 6 160.00 +-+++--+--- 7 576.00 -++++-+--++++ 8 2560.00 +-+++---+--+--- 9 12288.00 --+---+++-++-+++- 10 73728.00 ++-++-+++---+--+--- 11 327680.00 +-++++--+++-+-----+-- 12 2097152.00 +-++----+---+-++----+-- 13 14929920.00 -+-+++++-+++--+-+++++-+++ 14 68853760.00 +--+-+-----+++-++-+-+++++--
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