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SingularValueDecomposition

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  SingularValueDecomposition John Snyder <jsnyder@wi.rr.com> - 2011-06-18 21:45 +0000

#3178 — SingularValueDecomposition

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.

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