<?xml version="1.0" encoding="ISO-8859-1" standalone="no"?>
<!DOCTYPE MAN SYSTEM "../../manrev.dtd">
<MAN>
  <LANGUAGE>eng</LANGUAGE>
  <TITLE>eigenmarkov</TITLE>
  <TYPE>Scilab Function</TYPE>
  <DATE>April 1993</DATE>
  <SHORT_DESCRIPTION name="eigenmarkov"> normalized left and right Markov eigenvectors</SHORT_DESCRIPTION>
  <CALLING_SEQUENCE>
    <CALLING_SEQUENCE_ITEM>[M,Q]=eigenmarkov(P)  </CALLING_SEQUENCE_ITEM>
  </CALLING_SEQUENCE>
  <PARAM>
    <PARAM_INDENT>
      <PARAM_ITEM>
        <PARAM_NAME>P</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: real N x N Markov matrix. Sum of entries in each row should add to one.</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>M</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: real matrix with N columns.</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>Q</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: real matrix with N rows.</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
    </PARAM_INDENT>
  </PARAM>
  <DESCRIPTION>
    <P>
    Returns normalized left and right eigenvectors
    associated with the eigenvalue 1 of the Markov transition matrix P.
    If the multiplicity of this eigenvalue is m and P
    is N x N, M is a m x N matrix and Q a N x m matrix.
    M(k,:) is the probability distribution vector associated with the kth
    ergodic set (recurrent class). M(k,x) is zero if x is not in the
    k-th recurrent class.
    Q(x,k) is the probability to end in the k-th recurrent class starting
    from x. If <VERB>P^k</VERB> converges for large <VERB>k</VERB> (no eigenvalues on the
    unit circle except 1), then the limit is <VERB>Q*M</VERB> (eigenprojection).</P>
  </DESCRIPTION>
  <EXAMPLE>
<![CDATA[
//P has two recurrent classes (with 2 and 1 states) 2 transient states
P=genmarkov([2,1],2) 
[M,Q]=eigenmarkov(P);
P*Q-Q
Q*M-P^20
 ]]>
  </EXAMPLE>
  <SEE_ALSO>
    <SEE_ALSO_ITEM>
      <LINK>genmarkov</LINK>
    </SEE_ALSO_ITEM>
    <SEE_ALSO_ITEM>
      <LINK>classmarkov</LINK>
    </SEE_ALSO_ITEM>
  </SEE_ALSO>
</MAN>
