Re: General 3-state stochastic matrix

*To*: mathgroup at smc.vnet.net*Subject*: [mg58227] Re: General 3-state stochastic matrix*From*: Maxim <ab_def at prontomail.com>*Date*: Thu, 23 Jun 2005 05:34:17 -0400 (EDT)*References*: <d98p16$eeo$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

On Tue, 21 Jun 2005 10:09:42 +0000 (UTC), 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: > ... > > 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 > Actually, T is a transpose of a stochastic matrix. Mathematica will most likely have problems with this limit, because even in cases like Limit[(a/2)^n, n -> Infinity, Assumptions -> a > 2] Limit is just left unevaluated. One method is to use the properties of X = T^Infinity: In[1]:= T = {{1 - a2 - a3, a2, a3}, {b1, 1 - b1 - b3, b3}, {c1, c2, 1 - c1 - c2}}; X = Array[x, {3, 3}]; In[3]:= sol = First@ Solve[T.X == X.T == X, Variables@ X]; sol2 = First@ Solve[Eigenvalues@ X == {0, 0, 1} /. sol, Variables@ X]; X /. sol /. sol2 // Simplify Out[5]= {{(b3*c1 + b1*(c1 + c2))/(b1*c1 + b3*c1 + b1*c2 + a3*(b1 + b3 + c2) + a2*(b3 + c1 + c2)), (a3*c2 + a2*(c1 + c2))/(b1*c1 + b3*c1 + b1*c2 + a3*(b1 + b3 + c2) + a2*(b3 + c1 + c2)), (a2*b3 + a3*(b1 + b3))/(b1*c1 + b3*c1 + b1*c2 + a3*(b1 + b3 + c2) + a2*(b3 + c1 + c2))}, {(b3*c1 + b1*(c1 + c2))/(b1*c1 + b3*c1 + b1*c2 + a3*(b1 + b3 + c2) + a2*(b3 + c1 + c2)), (a3*c2 + a2*(c1 + c2))/(b1*c1 + b3*c1 + b1*c2 + a3*(b1 + b3 + c2) + a2*(b3 + c1 + c2)), (a2*b3 + a3*(b1 + b3))/(b1*c1 + b3*c1 + b1*c2 + a3*(b1 + b3 + c2) + a2*(b3 + c1 + c2))}, {(b3*c1 + b1*(c1 + c2))/(b1*c1 + b3*c1 + b1*c2 + a3*(b1 + b3 + c2) + a2*(b3 + c1 + c2)), (a3*c2 + a2*(c1 + c2))/(b1*c1 + b3*c1 + b1*c2 + a3*(b1 + b3 + c2) + a2*(b3 + c1 + c2)), (a2*b3 + a3*(b1 + b3))/(b1*c1 + b3*c1 + b1*c2 + a3*(b1 + b3 + c2) + a2*(b3 + c1 + c2))}} Or In[6]:= Module[{s, j}, {s, j} = JordanDecomposition[T]; s.Replace[j, {1 -> 1, _ -> 0}, {2}].Inverse[s] == %5 // Simplify ] Out[6]= True I think this will hold for degenerate cases too, as an appropriate limit. Maxim Rytin m.r at inbox.ru