Re: [math-fun] Geometry question
Nice one. Interestingly, a paper published only this year proves that if an NxN Hadamard matrices is also a circulant -- like Rich's matrix -- then N is 1 or 4. See < http://blms.oxfordjournals.org/content/early/2011/01/24/blms.bdq112.full.pdf >. --Dan ----- Rich wrote: << How about recasting as - + + + + - + + + + - + + + + - for "obvious" symmetry? To go from your reverse-quaternion-sign matrix to mine, negate the first column and the last three rows; then swap rows 2 & 4.
Sometimes the brain has a mind of its own.
On the other hand, there exist Hadamard matrices of all orders 2^k which are "extended cyclic" --- that is, with an initial row and column of + except for the diagonal, and the remainder of the matrix cyclic (and indeed symmetric) --- as in Dan's original layout. These are just the Reed-Muller codes. WFL On 7/23/11, Dan Asimov <dasimov@earthlink.net> wrote:
Nice one.
Interestingly, a paper published only this year proves that if an NxN Hadamard matrices is also a circulant -- like Rich's matrix -- then N is 1 or 4.
See < http://blms.oxfordjournals.org/content/early/2011/01/24/blms.bdq112.full.pdf
.
--Dan -----
Rich wrote:
<< How about recasting as - + + + + - + + + + - + + + + - for "obvious" symmetry? To go from your reverse-quaternion-sign matrix to mine, negate the first column and the last three rows; then swap rows 2 & 4.
Sometimes the brain has a mind of its own.
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There appears to be a flaw in the proof: http://people.math.sfu.ca/~jed/Papers/Craigen%20Jedwab.%20Circulant%20Hadama... Veit On Jul 22, 2011, at 8:38 PM, Dan Asimov wrote:
Nice one.
Interestingly, a paper published only this year proves that if an NxN Hadamard matrices is also a circulant -- like Rich's matrix -- then N is 1 or 4.
See < http://blms.oxfordjournals.org/content/early/2011/01/24/blms.bdq112.full.pdf >.
--Dan -----
Rich wrote:
<< How about recasting as - + + + + - + + + + - + + + + - for "obvious" symmetry? To go from your reverse-quaternion-sign matrix to mine, negate the first column and the last three rows; then swap rows 2 & 4.
Sometimes the brain has a mind of its own.
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