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!

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