rotation matrix

[bhelmes]

Originally Posted by Dr Sardonicus

No, sir. If M is a 2x2 matrix, scalar multiplication of M by k produces a matrix with determinant k^2*det(M). Thus, if M is a nonsingular 2x2 matrix, det((1/det(M))M) is 1/det(M).

If M is 2x2 and det(M) is not a square, no scalar multiple of M will have determinant 1.

A peaceful, early morning, especially for Dr Sardonicus,

let: p=31, u1=2, v1=3; so that the norm (u1,v1)=u1²+v1²=13=12⁻¹ mod 31 and 13²=20⁻¹ mod 31

Is it possible from linear algebra to calculate one belonging rotation matrix from this vector ?

The calculated target is:
13*
(27,2)* (2)=(5)
(-2,27) (3)=(9)

13*
(14,17)* (2)=(4)
(-17,14) (3)=(11)

13*
(24,8)* (2)=(6)
(-8,24) (3)=(15)

http://localhost/devalco/unit_circle/system_tangens.php

(the red coloured boxes right, all calculations checked and it seems to be all right.)


My first try:
Let p=31

Let M1=13*M1*=
13*
(a*, b*)
(-b*, a*)

1. with det (M1)=1=det (13² * det (M1*)) so that det (M1*)=20 mod 31, therefore a*²+b*²=20
2. with M1*(u1,v1)=(u2,v2) with norm (u1,v1)=(u2,v2)=u1²+v1²=u2²+v2²=13 mod p

so that

13*
(a*, b*) = (u2)
(-b*, a*) (v2)

This is more a fragment and should point in one direction, and as it is too late for me,
I hope that some one could finish the calculation.

[Dr Sardonicus]

It appears you are trying to solve

R[a;b] = [c;d]

where

R = [x,y;-y,x].

Writing as a system of linear equations,

x*a + y*b = c
b*x - a*y = d

which may be rewritten

[a,b;b,-a][x;y]=[c;d]

which is solvable if a^2 + b^2 ≠ 0.

EDIT: Feeding the formula to Pari-GP produces the same values you got:

? matsolve([Mod(2,31),Mod(3,31);Mod(3,31),Mod(-2,31)],Mod(1/13,31)*[Mod(5,31);Mod(9,31)])
%1 = 
[Mod(27, 31)]

[Mod(2, 31)]

? matsolve([Mod(2,31),Mod(3,31);Mod(3,31),Mod(-2,31)],Mod(1/13,31)*[Mod(4,31);Mod(11,31)])
%2 = 
[Mod(14, 31)]

[Mod(17, 31)]

? matsolve([Mod(2,31),Mod(3,31);Mod(3,31),Mod(-2,31)],Mod(1/13,31)*[Mod(6,31);Mod(15,31)])
%3 = 
[Mod(24, 31)]

[Mod(8, 31)]

[bhelmes]

A peaceful and pleasant night for you,

If Mp is a Mersenne number with exponent p
I can predict the following rotation matrix

M=(Mp-2^[(p-1)/2];-2^[(p-1)/2)
(2^[(p-1)/2]; Mp-2^[(p-1)/2])
with det (M)=1

Example: Mp=31 p=5

M=(27, -4)
(4, 27)
with det (M)=16+16=1 mod 31.

This rotation matrix is not a result of two vectors (x1,y1), (x2, y2) with the same norm x1²+y²=x2²+y2²=n mod Mp,
which needed to be found in advance,

but it is completely predictive.

I did not understand, why this matrix occurs one time for non quadratic residues and one time for quadratic residues:

(27,4)* (2)=(4)
(-4,27) (3)=(11)
with 22+32 = 13 and jac (13, 31)=-1

(27,4)* (5)=(8)
(-4,27) (7)=(14)
52+72 = 12 and jac (12, 31)=-1

(27,27)* (8)=(1)
(-27,27) (15)=(3)
82+152 = 10 and jac (10, 31)=1

Is there a mathematical explication for this phenomenon ?

More examples and as usual a web interface under http://devalco.de/unit_circle/system_tangens.php

Enjoy the predictable rotation matrix,

[Dr Sardonicus]

If 2*x^2 == 1 (mod N) and M = x[-1,-1;1,-1] we have det(M) = 2*x^2 == 1 (mod N) and M^2 = x^2[0,2;-2,0] == [0,1;-1,0] (mod N), so that M^8 == [1,0;0,1] (mod N).

One can take N = 2^n - 1 with n > 1 odd, and x = 2^((n-1)/2). However, suitable x will exist for any N composed entirely of primes congruent to 1 or 7 (mod 8).

[bhelmes]

There should be more peace in the world !

The pythagorean triples can be used to make a surjective mapping to the rotation matrices concerning p

if x²+y²=z² | : z²
(x/z)²+(y/z)²=1 which is the determinant of the rotation matrix
a=x/z mod p; b=y/z mod p

M=
(a, b)
(-b, a)

Question:

if M*(n, m)=(u, v), then n²+m²=u²+v² mod p and

if jacobi (n²+m², p)=-1 resp. n²+m² a non quadratic residue concerning p and p = 3 mod 4

is there a chance that [(n+mi)/(u,vi)]^r=1 mod p [r=(p+1)/2] and r maximal and not a right divisor of (p+1)/2 ?

Or in other words: Is it possible to say anything about the group order of the resulting vector
which is the divisor of the two vectors connected by the rotation matrix ?

A peaceful 1. advent


Of course a wonderful php scripting image with more infos you will need:
http://devalco.de/unit_circle/system_tangens.php

[bhelmes]

A peaceful and beautiful night for you, where ever you stay,

if you regard primes with p>3 as circles,
where the points on the circles are composed by the following different "norms":

1. euclidean norm with x²+y² = (x+yI)(x-yI)= 0 mod p
2. 2x²-y² = (sqrt(2)x+y)(sqrt(2)x-y)=0 mod p
3. 2x²+y²= (sqrt(2)x+yI)(sqrt(2)x-yI)=0 mod p

I think there has to exist several "rotation matrices" for 2. and 3.
which "move" the belonging points on the circle for a prime p.

Is there a way to derive and compose these "rotation matrices",
especially for 2. which may be important for the factorization of Mersenne numbers ?

And sorry if the description is not mathematical perfect formulated.

Mathematical feedback is welcome.