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<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org" xml:lang="en" xml:id="eigenmarkov">
    <refnamediv>
        <refname>eigenmarkov</refname>
        <refpurpose>normalized left and right Markov eigenvectors</refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Syntax</title>
        <synopsis>[M,Q]=eigenmarkov(P)</synopsis>
    </refsynopsisdiv>
    <refsection>
        <title>Arguments</title>
        <variablelist>
            <varlistentry>
                <term>P</term>
                <listitem>
                    <para>real N x N Markov matrix. Sum of entries in each row should add to one.</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>M</term>
                <listitem>
                    <para>real matrix with N columns.</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>Q</term>
                <listitem>
                    <para>real matrix with N rows.</para>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection>
        <title>Description</title>
        <para>
            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 <literal>P^k</literal> converges for large <literal>k</literal> (no eigenvalues on the
            unit circle except 1), then the limit is <literal>Q*M</literal> (eigenprojection).
        </para>
    </refsection>
    <refsection>
        <title>Examples</title>
        <programlisting role="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
 ]]></programlisting>
    </refsection>
    <refsection role="see also">
        <title>See also</title>
        <simplelist type="inline">
            <member>
                <link linkend="genmarkov">genmarkov</link>
            </member>
            <member>
                <link linkend="classmarkov">classmarkov</link>
            </member>
        </simplelist>
    </refsection>
</refentry>
