Before going on with a new idea I suggest a look at "Matrix Moves" in this thread.
Supposing we have the stochastic matrix
The central problems of Linear Algebra are the solution of a simultaneous set of linear equations or Ax = b and determining the eigenvalues of a matrix.
The eigenvalues are usually computed using a computer and we will not break with tradition, they are
There is a little theorem that says if a square matrix has distinct eigenvalues then it is diagonalizable. So this one is diagonalizable.
To do it we need the Eigenvectors of A:
To check whether we have diagonalized it we plug in to
Okay, so what? The useful fact is that to get A^k we only now need the following matrix equation.
Now D^k is easy to get because to raise a matrix with just diagonal elements like D to the kth power you just take each element and raise it to the kth power.
So if we wanted A^10 we would compute
And we are done!
In mathematics, you don't understand things. You just get used to them.
I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.