a "yellow" generally orthogonal Hadamard matrix type
- To: mathgroup at smc.vnet.net
- Subject: [mg101034] a "yellow" generally orthogonal Hadamard matrix type
- From: Roger Bagula <rlbagula at sbcglobal.net>
- Date: Sun, 21 Jun 2009 07:07:44 -0400 (EDT)
Yesterday, by using my ideas of product self-similat matrices I got: The matrix products work: Hadamard ( boson like) h2 = {{1, 1}, {1, -1}} Triangular Hadamard ( lepton like) ht2 = {{1, 0}, {1, -1}} Product one way: h = h2.ht2 {{2, -1}, {0, 1}} Product the other way: h1 = ht2.Transpose[h2] {{1, 1}, {0, 2}} h.h1 {{2, 0}, {0, 2}} I found that hy=ht.h2.ht was better! The matrix type {2,-1}, {2,-2}} self-similar matrices gave me this idea: If the original Haramard were "white", this would be yellow? {{a,-1} {a,-a}} The square is generally:a>=2, Integer {{-a + a2, 0}, {0, -a+ a2}} Table[n2 - n, {n, 2, 10}] {2, 6, 12, 20, 30, 42, 56, 72, 90} Since it used itself instead of the transpose for orthogonality, it is like the triangular Hadamard (not the original boson sort). That it is orthogonal and works as a self-similar matrix I did using a=Prime[n]: {{2, -1}, {2, -2}}, {{9, -3, -3, 1}, {9, -9, -3, 3}, {9, -3, -9, 3}, {9, -9, -9, 9}}, {{125, -25, -25, 5, -25, 5, 5, -1}, {125, -125, -25,25, -25, 25, 5, -5}, {125, -25, -125, 25, -25, 5, 25, -5}, {125, -125, -125, 125, -25, 25, 25, -25}, {125, -25, -25, 5, -125, 25, 25, -5}, {125, -125, -25, 25, -125, 125, 25, -25}, {125, -25, -125, 25, -125, 25, 125, -25}, {125, -125, -125, 125, -125, 125, 125, -125}} The squares are: {{2, 0}, {0, 2}}, {{36, 0, 0, 0}, {0, 36, 0, 0}, {0, 0, 36, 0}, {0, 0, 0, 36}}, {{8000, 0, 0, 0, 0, 0, 0, 0}, {0, 8000, 0, 0, 0,0, 0, 0}, {0, 0, 8000, 0, 0, 0,0, 0}, {0, 0, 0, 8000, 0, 0, 0, 0}, {0, 0, 0, 0, 8000, 0,0, 0}, {0, 0, 0, 0, 0, 8000, 0, 0}, {0, 0, 0, 0, 0, 0, 8000, 0}, {0, 0, 0, 0, 0, 0, 0, 8000}} So my ideas about product sets hasn't been without some results. http://vids.myspace.com/index.cfm?fuseaction=vids.individual&videoid=59286373 I couldn't get it up to webMathematica so I tried this. Mathematica for animation: Clear[HadamardMatrix, HadamardMatrix1] MatrixJoinH[A_, B_] := Transpose[Join[Transpose[A], Transpose[B]]] KroneckerProduct[M_, N_] := Module[{M1, N1, LM, LN, N2}, M1 = M; N1 = N; LM = Length[M1]; LN = Length[N1]; Do[M1[[i, j]] = M1[[i, j]]N1, {i, 1, LM}, {j, 1, LM}]; Do[M1[[i, 1]] = MatrixJoinH[M1[[i, 1]], M1[[i, j]]], {j, 2, LM}, { i, 1, LM}]; N2 = {}; Do[AppendTo[N2, M1[[i, 1]]], {i, 1, LM}]; N2 = Flatten[N2]; Partition[N2, LM*LN, LM*LN]] HadamardMatrix[0] := {{1}} HadamardMatrix[2] := {{2, -1}, {2, -2}} HadamardMatrix[n_] := Module[{m}, m = {{2, -1}, {2, -2}}; KroneckerProduct[m, HadamardMatrix[n/2]]] HadamardMatrix1[n_] := If[Mod[n, 2] == 0 && IntegerQ[ Log[2, n]], HadamardMatrix[n], Table[HadamardMatrix[2^(Floor[ Log[2, n]] + 1)][[i, j]], {i, n}, {j, n}]] a = Table[ListDensityPlot[HadamardMatrix1[n], Mesh -> False, AspectRatio -> Automatic, ColorFunction -> Hue, Axes -> False, Frame -> False], { n, 2, 32}] Respectfully, Roger L. Bagula 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :http://www.geocities.com/rlbagulatftn/Index.html alternative email: rlbagula at sbcglobal.net