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Author Topic: A Gray code Hadamard matrix self-similarity  (Read 2623 times)
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tftn
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« on: November 02, 2006, 08:42:24 PM »

A Gray code Hadamard matrix self-similarity zoom movie:

http://www.mathematica-users.org/mediawiki/images/2/25/hadamard_64_zoom.avi <http://www.mathematica-users.org/mediawiki/images/2/25/hadamard_64_zoom.avi>

Herev are two together ( bigger file):
http://www.mathematica-users.org/mediawiki/images/3/38/hadamard64_32.avi <http://www.mathematica-users.org/mediawiki/images/3/38/hadamard64_32.avi>

New movie:
http://www.mathematica-users.org/mediawiki/images/4/4d/had_P_GC_64.avi

My observation is that these look like Wolfram's CA's.

Documentation of the math:
%I A000001
%S A000001 1, 1, -1, 1, -2, 1, 0, -2, 3, -1, 1, 0, -2, 0, 1, 0, -2, -1, 3, 1, -1, 0, 0,
-3, 6, -2, -2, 1, 0, 2, -9, 15, -11, 3, 1, -1, 1, -4, 2, 6, -1, -6, -1, 2, 1,
0, -2, 7, -1, -11, -3, 8, 4, -1, -1, 0, 0, -3, -6, 4, 18, -9, -2, -3, 0, 1,
0, 0, 0, -4, 3, 19, -29, 11, -2, 2, 1, -1, 0, 0, 0, 0, 4, 0, -25, 16, 26,
-20, -4, 2, 1, 0, 0, 0, -4, 11, 7, -63, 63, 8, -34, 15, 1, -3, -1, 0, 0, -3,
12, -23, 46, -123, 176, -74, -64, 84, -32, -4, 4, 1, 0, -2, 3, 9, 43, -263,
397, -259, 119, 17, -134, 32, 56, -8, -9, -1, 1, 4, -18, -36, 157, -50, -220,
78, 226, -64, -141, 20, 56, -2, -12, 0, 1, 0, -2, -9, 31, 85, -265, -8, 382,
6, -368, -14, 204, 23, -69, -9, 13, 1, -1, 0, 0, -3, 30, -92, 6, 277, -30,
-774, 802, -208, 100, -129, -22, 38, 16, -10, -2, 1, 0, 0, 0, -4, 11, 95,
-41, -341, 210, 666, -1013, 445, -49, 39, 9, -17, -21, 9, 3, -1, 0, 0, 0, 0,
4, -8, -107, -16, 435, -50, -940, 694, 473, -630, 136, -68, 89, 6, -19, 0, 1
%N A000001 n th level Hadamard matrices for Pascal type binary are inversted and multiplied
by the corresponding n th level Gray code Hadamard ( some of which matrices have determinant zero): the resulting matrix is processed to give a triangular
sequence.
%C A000001 Matrices:
1by1
{{1}}
2by2
{{1, 0},
{0, 1}}
3by3
{{1, -1, -1},
{0, 1, 1},
{0, 1, 1}}
4by4
{{1, 1, 0, 0},
{0, -1, 0, 0},
{0, -1, 0, 1},
{0, 2, 1, 0}}
5by5
{{1, 1, -1, -1, -1},
{0, -1, 0, 0, 0},
{0, -1, 0, 1, 1},
{0, 2, 1, 0, 0},
{0, 0, 1, 1, 1}}
6by6
{{1, 1, 0, -1, -1, 0},
{0, -1, -1, 0, 0, -1},
{0, -1, 0, 1, 1, 0},
{0, 2, 1, 0, 0, 1},
{0, 0, 0, 1, 1, 0}.
{0, 0, 1, 0, 0, 1}}
They don't get interesting until 4by4!
%F A000001 x(i,j)=a(i,j)^(-1).b(i,j)
p(n,x)=CharacteristicPolyynomial(x(i,j))
p(n,x)->t(n,m)
%e A000001 Triangular sequence:
{1},
{1, -1},
{1, -2, 1},
{0, -2, 3, -1},
{1, 0, -2, 0, 1},
{0, -2, -1, 3,1, -1},
{0, 0, -3, 6, -2, -2, 1},
{0, 2, -9, 15, -11,3, 1, -1},
{1, -4, 2, 6, -1, -6, -1, 2, 1},
{0, -2,7, -1, -11, -3, 8, 4, -1, -1},
{0, 0, -3, -6, 4, 18, -9, -2, -3, 0, 1}
Polynomials:
1,
1 - x,
1 - 2 x + x2,
0 -2x + 3x2 - x3,
1 +0x - 2x2 + x4,
0-2x - x2 + 3 x3 + x4 - x5,
0+0x +3x2 + 6 x3 - 2 x4 - 2 x5 + x6,
0+ 2x - 9 x2 + 15x3 - 11 x4 + 3 x5 + x6 -x7,
1 - 4 x + 2x2 + 6x3 - x4 - 6 x5 - x6 +  2 x7 + x8
%t A000001 c[i_, k_] := Floor[Mod[i/2^k, 2]];
b[i_, k_] := If[c[i, k] == 0 && c[
i, k + 1] == 0, 0, If[c[i, k] == 1 && c[i, k + 1] == 1, 0, 1]];
An[d_] := Table[If[Sum[b[n, k]*b[m, k], {k,
   0, d - 1}] == 0, 1, 0], {n, 0, d - 1}, {m, 0, d - 1}];
Bn[d_] := Table[If[Sum[c[n, k]*c[
m, k], {k, 0, d - 1}] == 0, 1, 0], {n, 0, d - 1}, {m, 0, d - 1}];
Xn[d_] := MatrixPower[Bn[d], -1].An[d];
a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[Xn[d], x], x],
{d, 1, 20}]];
Flatten[%]
%Y A000001 Cf. A122944, A121801,A122947
%O A000001 1
%K A000001 ,nonn,
%A A000001 Gary Adamson and Roger Bagula (rlbagula@sbcglobal.net), Oct 26 2006
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