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
