General 3-state stochastic matrix

*To*: mathgroup at smc.vnet.net*Subject*: [mg58164] General 3-state stochastic matrix*From*: Virgil Stokes <virgil.stokes at it.uu.se>*Date*: Tue, 21 Jun 2005 06:02:37 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

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: Clear[Î±2, Î±3, Î²1, Î²3, Î³1, Î³2] T = {{1 - Î±2 - Î±3, Î±2, Î±3}, {Î²1, 1 - Î²1 - Î²3, Î²3}, {Î³1, Î³2, 1 - Î³1 - Î³2}}; MatrixForm[T] xx = MatrixPower[T, n]; TimeUsed[] zz = Limit[xx, n -> â??, Assumptions -> 0 < Î±2 < 1 && 0 < Î±3 < 1 && 0 < Î±2 + Î±3 < 1 && 0 < Î²1 < 1 && 0 < Î²3 < 1 && 0 < Î²1 + Î²3 < 1 && 0 < Î³1 < 1 && 0 < Î³2 < 1 && 0 < Î³1 + Î³2 < 1] // FullSimplify; TimeUsed[] MatrixForm[zz] 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. Note, for a general 2-state stochastic matrix, the above approach works fine. --Thanks, V. Stokes