%0 Journal Article
%T Left Eigenvector of a Stochastic Matrix
%A Sylvain Lavalle∩e
%J Advances in Pure Mathematics
%P 105-117
%@ 2160-0384
%D 2011
%I Scientific Research Publishing
%R 10.4236/apm.2011.14023
%X We determine the left eigenvector of a stochastic matrix  &lt;i&gt;M&lt;/i&gt; associated to the eigenvalue 1 in the commutative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenvalue 0 is (&lt;i&gt;N&lt;/i&gt;&lt;sub&gt;1&lt;/sub&gt;,&lt;i&gt;N&lt;sub&gt;n&lt;/sub&gt;&lt;/i&gt;) , where &lt;i&gt;N&lt;/i&gt;&lt;sub&gt;i&lt;/sub&gt;  is the   &lt;i&gt;i&lt;/i&gt;每&lt;i&gt;th&lt;/i&gt; iprincipal minor of &lt;i&gt;N&lt;/i&gt;=&lt;i&gt;M&lt;/i&gt;每&lt;i&gt;I&lt;sub&gt;n&lt;/sub&gt;&lt;/i&gt; , where  &lt;i&gt;I&lt;sub&gt;n&lt;/sub&gt;&lt;/i&gt; is the identity matrix of dimension  &lt;i&gt;n&lt;/i&gt;. In the noncommutative case, this eigenvector is (&lt;i&gt;P&lt;/i&gt;&lt;sub&gt;1&lt;/sub&gt;&lt;sup&gt;-1&lt;/sup&gt;,&lt;i&gt;P&lt;/i&gt;&lt;sub&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sub&gt;&lt;sup&gt;-1&lt;/sup&gt;)  , where  &lt;i&gt;P&lt;sub&gt;i&lt;/sub&gt;&lt;/i&gt; is the sum in  Q▲汐&lt;sub&gt;&lt;i&gt;ij&lt;/i&gt;&lt;/sub&gt;◎ of the corresponding labels of nonempty paths starting from  &lt;i&gt;i&lt;/i&gt; and not passing through &lt;i&gt;i&lt;/i&gt;   in the complete directed graph associated to  &lt;i&gt;M&lt;/i&gt;  .
%K Generic Stochastic Noncommutative Matrix
%K Commutative Matrix
%K Left Eigenvector Associated To The Eigenvalue 1
%K Skew Field
%K Automata
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=6483