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:
h2 = {{1, 1},
{1, -1}}
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:

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[2] := {{2, -1}, {2, -2}}

m = {{2, -1}, {2, -2}};

HadamardMatrix1[n_] := If[Mod[n, 2] == 0 && IntegerQ[

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
:http://www.geocities.com/rlbagulatftn/Index.html
alternative email: rlbagula at sbcglobal.net

```

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