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#1 2013-01-26 10:38:41

bobbym
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Diagonalizing a matrix.

Hi;

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 have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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