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From: fuji on 1 Aug 2010 09:11
Theorem (All system is on 2 dimensions.)
There are the number of n special linear transformation matrices,
Ai=(ai,bi,ci,di) [(11，12，21，22)], det Ai=1 (special), i=1 to n. From
any starting point P1, transforming one by one by Ai,i.e,
P2=A1*P1,P3=A2*P2,..,Pi+1=Ai*Pi,i=1 to n, comes full closed line and
ends with the starting point P1=An*Pn.
The sequence of transformation can be exchanged freely as Ai*Ai
+1*Pi=Ai*Ai+1*Pi (interchangeable symmetry), but the starting and
ending point P1 is fixed. The necessary and sufficient condition to
stand up this way is that one function f(x,y) is invariant for all
the transformation matrices Ai,i=1 to n.
f(x,y)=k*x*x-y*y+h*x*y , where k=ci/bi, h=(di-ai)/bi, i=1 to n.
and at this time, we get ΠAi=Eigen matrix, i=1 to n(any order will
do), and f(P1)=f(P2)=...=f(Pn).
Corollary
On the sprcial linear transformation Ai, when ai=di, the function
f(x,y) is
f(x,y)=k*x*x-y*y, k=ci/bi, i=1 to n
and when αi,βi are the eigen value of Ai, following equation is
obtained.
Παi=Πβi=Π(ai+sqrt(k)*bi)=Π(ai-sqrt(k)*bi)=1, i=1 to n

This theorem is related with the Theory oh Relativity.
To be continued.

H.Fujimori
From: fuji on 2 Aug 2010 08:51
On 8月1日, 午後10:11, fuji <r...(a)r2.dion.ne.jp> wrote:
> Theorem      (All system is on 2 dimensions.)
> There are the number of n special linear transformation matrices,
> Ai=(ai,bi,ci,di) [(11，12，21，22)], det Ai=1 (special), i=1 to n. From
> any starting point P1, transforming one by one by Ai,i.e,
> P2=A1*P1,P3=A2*P2,..,Pi+1=Ai*Pi,i=1 to n, comes full closed line and
> ends with the starting point P1=An*Pn.
> The sequence of transformation can be exchanged freely as Ai*Ai
> +1*Pi=Ai*Ai+1*Pi (interchangeable symmetry), but the starting and
> ending point P1 is fixed. The necessary and sufficient condition to
> stand up this way is that  one function f(x,y) is invariant for all
> the transformation matrices Ai,i=1 to n.
>  f(x,y)=k*x*x-y*y+h*x*y   , where k=ci/bi, h=(di-ai)/bi, i=1 to n.
> and at this time, we get ΠAi=Eigen matrix, i=1 to n(any order will
> do), and  f(P1)=f(P2)=...=f(Pn).
> Corollary
> On the sprcial linear transformation Ai, when ai=di, the function
> f(x,y) is
>  f(x,y)=k*x*x-y*y,  k=ci/bi, i=1 to n
> and when αi,βi are the eigen value of Ai, following equation is
> obtained.
>  Παi=Πβi=Π(ai+sqrt(k)*bi)=Π(ai-sqrt(k)*bi)=1, i=1 to n
>
> This theorem is related with the Theory oh Relativity.
> To be continued.
>
> H.Fujimori

There was a mistake in the former article.
wrong >The sequence of transformation can be exchanged freely as Ai*Ai
+1*Pi=Ai*Ai+1*Pi
correction ---> Ai+1*Ai*Pi=Ai*Ai+1*Pi

The last equation is explained that the product of Ai(s) are
equivalent not only in due order but also in inverse order only if
transformation circulates in circumference. This equation can be
separated into three types by the sign of constnt k.
(i)k<0, ellipse type
When Ai=(ai,bi,ci,ai)=(cosθi,-sinθi,sinθi,cosθi), Ai reveals
coordinate revolution transformation by θi, and k=ci/bi=-1. Defining
j=sqrt(-1) and using Euler relation equation exp(j*θ)=cosθ+j*sinθ, the
last equation comes to
Π(cosθi-j*sinθi)=Π(cosθi+j*sinθi)=1, i=1 to n
Therefoe exp(jΣ-θi)=exp(jΣθi)=1, thus Σθi=2mπ. This is a very natural
conclusion.
(ii)k=0, line type
When ai=di=1, and defining -vi=bi/ai, we get Galilean transformation,
v1+v2+...vn=0
(iii)k>0, hyperbola type
Because k>0 means ai>1 (det Ai=1), and defining -vi=bi/ai, then Ai
becomes Lorentz transformation matrix. When n=3, the last equation is
-v3=(v1+v2)/(1+k*v1*v2) (velocity addition)
All I have discussed is in the world of mathematics, and velocity
addition equation resulted. This implys that we can prove why the
light speed is invariant.

H.Fujimori
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