Re: matrix substitution
- To: mathgroup at smc.vnet.net
- Subject: [mg67920] Re: matrix substitution
- From: Roger Bagula <rlbagula at sbcglobal.net>
- Date: Fri, 14 Jul 2006 02:11:14 -0400 (EDT)
- References: <e665nv$n43$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
References questioned by Paul Abbott in this thread:
volume 3, page 336 , Mathematics ,its content, methods and meaning,MIT
press ,Mass. 1963
The Russian 2by2 matrix transform of complex numbers:
a+I*b--> {{a,b},{-b,a}}
1->{{1,0},{0,1}}
I->{{0,1},{-1,0}}
0->{0,0},{0,0}}
The Russian Non-Euclidean transform: ( page 120 volume 3)
x'=(x+a)/(1+a*x)
y'=(y*Sqrt[1-a^2)/(1+a*x)
Roger Bagula wrote:
>In certain complex 2by2 groups there
>are equivalent 4by4 representations
>whicg you can get by the substitutiuon:
>
>1->{{1,0},{0,1}}
>I->{{0,1},{-1,0}}
>An example;
>I used the code :
>
>a = {{1, 0}, {0, 1}}
>b = {{0, 1}, {-1, 0}}
>c = {{0, 0}, {0, 0}}
>i = MatrixForm[{{c, a}, {-a, c}}]
>j = MatrixForm[{{c, b}, {b, c}}]
>k = MatrixForm[{{b, c}, {c, -b}}]
>e = MatrixForm[{{a, c}, {c, a}}]
>q[t_, x_, y_, z_] = ExpandAll[e*t + x*i + j*y + k*z]
>
>to get:
>i = {{0, 0, 1, 0}, {0, 0, 0, 1}, {-1, 0, 0, 0}, {0, -1, 0, 0}}
>
>j = {{0, 0, 0, 1}, {0, 0, -1, 0}, {0, 1, 0, 0}, {-1, 0, 0, 0}}
>
>k = {{0, 1, 0, 0}, {-1, 0, 0, 0}, {0, 0, 0, -1}, {0, 0, 1, 0}}
>
>e = {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}
>( Try e^2-i^2-j^2-k^2)
>But I got the result by hand after the fact.
>There should be a way to do the derivation directly,
>but my skills aren't up to it, yet.
>
>My current problem is quotient group made from the golden mean
>characteristic:
>
>a=Table[x^n,{n,0,1}]
>b=Table[PolynomialMod[a[[n]]*a[[m]],x^2-x-1],{n,1,2},{m,1,2}]
>MatrixForm[b]
>
>The result is a quotient group for the golden men.
>Which gives a multipilcation table like:1->e, x->I
>{{e,I},{I,e+I}}
>which I can do the same sort of substitution on,
>but I'd like to have a more effective method!
>
>The benefit of the 4b4 matrices is that the result is in real numbers
>without and complex numbers.
>
>
>