Re: SingularValueDecomposition

• To: mathgroup at smc.vnet.net
• Subject: [mg119728] Re: SingularValueDecomposition
• From: Dana DeLouis <dana.del at gmail.com>
• Date: Sun, 19 Jun 2011 07:24:48 -0400 (EDT)

```...  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}

Hi.  Sorry I can't help, but as a side note, this converged in 7 loops
vs 100.

FixedPoint[Chop[#.#]&,m][[2]]
{0,0,0,0,0,0,0,0,0,0,0,0,0.809663,0.190337}

= = = = = = = = = =
Dana DeLouis

On Jun 18, 5:45 pm, John Snyder <jsny... at wi.rr.com> wrote:
> 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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