{"id":76,"date":"2020-12-06T14:51:37","date_gmt":"2020-12-06T14:51:37","guid":{"rendered":"http:\/\/quentin-duchemin.alwaysdata.net\/wiki\/?p=76"},"modified":"2025-08-08T12:37:44","modified_gmt":"2025-08-08T12:37:44","slug":"perron-frobenius-theorem","status":"publish","type":"post","link":"https:\/\/quentin-duchemin.alwaysdata.net\/wiki\/index.php\/2020\/12\/06\/perron-frobenius-theorem\/","title":{"rendered":"Perron Frobenius Theorem"},"content":{"rendered":"\n

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Perron Frobenius Theorem is a well-known algebra result that finds applications in a large span of fields of Mathematics. One can mention Markov chains, population growth (with the Leslie matrix model) or the famous PageRank algorithm. In this post we prove the Perron-Frobenius Theorem for stochastic matrices.<\/p>\n\n\n\n

Perron Frobenius Theorem<\/h2>\n\n\n\n

Let us consider some integer \"d\geq1\" and \"Q \in \mathcal M_d(\mathbb R_+)\" a stochastic matrix, meaning that for all \"i \in [d]\", \"\sum_j Q_{i,j}=1\". In the following, we will denote \"Q^n=\left( q_{i,j}^{(n)}\right)_{(i,j) \in [d]^2}\". We assume that \"Q\" is irreducible which means that \"\forall i,j \in [d], \quad \exists n \geq 0 \quad q_{i,j}^{(n)} >0.\" Then the period of the point \"i\" defined by \"p_i=\mathrm{pgcd}(\{n \geq 0 \; : \; q_{i,i}^{(n)}>0\}\" does not depend on \"i\" meaning that there exists some \"p \in \mathbb N\" such that \"p_i=p \quad \forall i \in [d]\". Then: <\/p>\n\n\n\n