Let $M = \left( m _ { i , j } \right)$ be a stochastic matrix of $\mathcal { M } _ { n } ( \mathbb { R } )$ and $\lambda$ an eigenvalue (real or complex) of $M$. We denote by $\left( u _ { 1 } , \ldots , u _ { n } \right)$ the components (real or complex), in the canonical basis, of an eigenvector $u$ associated with $\lambda$. Let $h \in \{ 1 , \ldots , n \}$ such that $\left| u _ { h } \right| = \max _ { 1 \leqslant i \leqslant n } \left| u _ { i } \right|$. Show that $\left| \lambda - m _ { h , h } \right| \leqslant 1 - m _ { h , h }$. Deduce that $| \lambda | \leqslant 1$.
Let $M = \left( m _ { i , j } \right)$ be a stochastic matrix of $\mathcal { M } _ { n } ( \mathbb { R } )$ and $\lambda$ an eigenvalue (real or complex) of $M$. We denote by $\left( u _ { 1 } , \ldots , u _ { n } \right)$ the components (real or complex), in the canonical basis, of an eigenvector $u$ associated with $\lambda$.
Let $h \in \{ 1 , \ldots , n \}$ such that $\left| u _ { h } \right| = \max _ { 1 \leqslant i \leqslant n } \left| u _ { i } \right|$. Show that $\left| \lambda - m _ { h , h } \right| \leqslant 1 - m _ { h , h }$. Deduce that $| \lambda | \leqslant 1$.