Sunday, December 4, 2016

The change depends on direction of the motion: Generating Eigenvalues from special matrices (CON)

In the reference with the article of “Can We Solve a Nonlinear Equation With Many Variables? (Con)” posted on link: : http://emfps.blogspot.co.uk/2016/10/can-we-solve-nonlinear-equation-with_19.html and following to article of "The change depends on direction of the motion: Generating Eigenvalues from special matrices" posted on link: https://emfps.blogspot.com/2016/06/the-change-depends-on-direction-of.html, here is another new theorem as follows:


Theorem: “If and only if there are three points on a sphere which are members of natural number, they make a matrix 3*3 where the eigenvalue of this matrix is easily calculated by using below 
formulas:

A =
 a11    a12    a13
a21     a22    a23
a31     a32    a33

λ1 = a11 - a12  
 
λ2 = a11+ a12 + a13

Example (1):

The points P1 (1, 2, 2) and P2 (2, 1, 2) and P3 (2, 2, 1) are on a sphere with radius of 3 (r =3), these points make matrix A where we can calculate eigenvalue of this matrix as follows:

A = 

1
2
2
2
1
2
2
2
1

λ1 = 1 – 2 = -1  

λ2 = 1+ 2 +2 = 5

Example (2):
The points P1 (8, 16, 16) and P2 (16, 8, 16) and P3 (16, 16, 8) are on a sphere with radius of 24 (r =24), these points make matrix A where we can calculate eigenvalue of this matrix as follows:

A = 

8
16
16
16
8
16
16
16
8

λ1 = 8 – 16 = -8  

λ2 = 8+ 16 +16 = 40

Example (3):
The points P1 (4, 8, 8) and P2 (8, 4, 8) and P3 (8, 8, 4) are on a sphere with radius of 12 (r =12), these points make matrix A where we can calculate eigenvalue of this matrix as follows:

A = 

4
8
8
8
4
8
8
8
4
λ1 = 4 – 8 = -4 

λ2 = 4+ 8 +8 = 20