SingularValueDecomposition

*To*: mathgroup at smc.vnet.net*Subject*: [mg119718] SingularValueDecomposition*From*: John Snyder <jsnyder at wi.rr.com>*Date*: Sat, 18 Jun 2011 06:22:36 -0400 (EDT)

I read Jon McLoone's recent post on the WolframBlog concerning the solution of the drunken sailor's walk problem by using a Markov chain transition probability matrix. He mentions that it may also be possible to solve the problem using the SingularValueDecomposition function, but he does not illustrate this. I am trying to figure out how this could be done. Here is a simple "toy" example. Assume that I have the following Markov chain transition probability matrix m where each row sums to 1: m={{1/4,1/4,0,1/4,0,0,0,0,0,0,0,0,1/4,0},{1/4,1/4,1/4,0,1/4,0,0,0,0,0,0,0,0,0},{0,1/4,1/4,0,0,1/4,0,0,0,0,0,0,1/4,0},{0,0,0,1/4,1/4,0,1/4,0,0,0,0,0,1/4,0},{0,0,0,1/4,1/4,1/4,0,1/4,0,0,0,0,0,0},{0,0,0,0,1/4,1/4,0,0,1/4,0,0,0,1/4,0},{0,0,0,0,0,0,1/4,1/4,0,1/4,0,0,1/4,0},{0,0,0,0,0,0,1/4,1/4,1/4,0,1/4,0,0,0},{0,0,0,0,0,0,0,1/4,1/4,0,0,1/4,1/4,0},{0,0,0,0,0,0,0,0,0,1/4,1/4,0,1/4,1/4},{0,0,0,0,0,0,0,0,0,1/4,1/4,1/4,0,1/4},{0,0,0,0,0,0,0,0,0,0,1/4,1/4,1/4,1/4},{0,0,0,0,0,0,0,0,0,0,0,0,1,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,1}}; Assuming that I start in the position 2 (column 2 out of 3, in the first of 4 rows) I want to find the so-called "fixed point", the ultimate state density function, as the number of steps goes to infinity. I know that I can do this numerically using MatrixPower as follows (here is 100 steps which appears to be more than enough in this case): In[19]:= MatrixPower[N[m],100][[2]]//Chop Out[19]= {0,0,0,0,0,0,0,0,0,0,0,0,0.809663,0.190337} I believe that it is also possible to get this same result by using the SingularValueDecomposition function, but I cannot figure out how to get this to work. Can someone please show me how to use SingularValueDecomposition to get the same answer to this question? I know there are other ways to solve this, but I am really interested in using SingularValueDecomposition in this case. Thanks.