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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