Cooling matrix

Suppose you have a grid of temperatures such as:

I found this matrix C:

The interesting thing about C is when you do this operation: 

(matrix multiplication)

For the example matrix I gave A, this gives: 

The sum of all of these entries is exactly the same as the sum of all the entries in matrix A! 

If you look at matrix A and the resultant matrix you can see that every cell gets closer to the average value of all the entries in the matrix, which is what you would expect from a grid of temperatures settling into an average temperature. 

Let's look at the bottom right cell in A, that starts off at 7 and goes to 4.875, another application of the matrix multiplication with C gives: 

7

4.875

4.1406

3.83

3.673

3.584

3.531

3.498

3.478

...

You can see it is is sort of exponentially decaying to the average temperature of 3.444