Here is a partial graph of a Collatz-Matrix equation of size [math]2\times 2[/math] if the default Collatz parameters up to [math]x=11[/math]
The only thing I notice is how the points on the x-axis have a sequential growth.
Here is how each point was received. So, for a [math]2\times 2[/math] matrix, a Collatz-Matrix equation produces 2 matrix solutions. Each matrix solution will produce a coordinate or point on the graph. To find the x coordinate of a point, you multiply the first diagonal elements. To get the y coordinate, you multiply the second diagonal elements.
Also, one can integrate Collatz-Matrix equations by doing the following.
The integration of this Collatz-Matrix equation can also be interpreted as this:
An interesting thing about the graph is the points on the x-axis is that they are separated by two times each Collatz number.
The coordinates of these points are:
(8,0)
(28, 0)
(60, 57)
(104, 0)
(160, 0)
(228, 0)
(308, 0)
(400, 0)
(504, 0)
(610, 0)
(748, 0)
And the difference between the x values is as follows:
8, 20, 32, 44, 56, 68, 80, 92, 104, 116
Where the difference between each of these differences is 12.
Now, if these values are divided by 2, then the following occurs:
4, 10, 16, 22, 28, 34, 40, 46, 52, 58
It becomes a sequence of Collatz numbers. This pattern would continue as the x value of the Collatz-Matrix equation increases in size.