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MathGroup Archive 2005

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Re: General 3-state stochastic matrix

  • To: mathgroup at smc.vnet.net
  • Subject: [mg58251] Re: General 3-state stochastic matrix
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Fri, 24 Jun 2005 03:28:41 -0400 (EDT)
  • Organization: The University of Western Australia
  • References: <d98p16$eeo$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <d98p16$eeo$1 at smc.vnet.net>,
 Virgil Stokes <virgil.stokes at it.uu.se> wrote:

> I have tried to find the limit (as n, the power of the matrix, goes to 
> infinity) for the general 3-state stochastic matrix using the following 
> code:

It was very difficult to read this posting due to the weird fonts. 

> However, it does not find a symbolic solution. I would appreciate it 
> greatly if someone else could look at this and see if they are able to 
> get a symbolic solution. Warning! this can take considerable CPU time.

Instead of computing the power of a matrix using MatrixPower and then 
taking the limit, a better approach is to diagonalize the matrix first, 
so that computing the power becomes trivial. Also, since the maximal 
eigenvalue of a stochastic matrix is unity, all entries in this diagonal 
matrix vanish, except for the one corresponding to unit eigenvalue. The 
Notebook

  http://physics.uwa.edu.au/pub/Mathematica/MathGroup/StochasticMatrices.nb

shows how to obtain the steady state matrix for an n-state stochastic 
matrix. 

Cheers,
Paul

-- 
Paul Abbott                                      Phone: +61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul
        http://InternationalMathematicaSymposium.org/IMS2005/


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